author | wenzelm |
Thu, 08 Dec 2022 11:24:43 +0100 | |
changeset 76598 | 9f97eda3fcf1 |
parent 73932 | fd21b4a93043 |
child 77362 | 1a6103f6ab0b |
permissions | -rw-r--r-- |
14706 | 1 |
(* Title: HOL/Algebra/Coset.thy |
68582 | 2 |
Authors: Florian Kammueller, L C Paulson, Stephan Hohe |
3 |
||
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
4 |
With additional contributions from Martin Baillon and Paulo Emílio de Vilhena. |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
5 |
*) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
6 |
|
35849 | 7 |
theory Coset |
8 |
imports Group |
|
9 |
begin |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
10 |
|
61382 | 11 |
section \<open>Cosets and Quotient Groups\<close> |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
12 |
|
35847 | 13 |
definition |
14963 | 14 |
r_coset :: "[_, 'a set, 'a] \<Rightarrow> 'a set" (infixl "#>\<index>" 60) |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
15 |
where "H #>\<^bsub>G\<^esub> a = (\<Union>h\<in>H. {h \<otimes>\<^bsub>G\<^esub> a})" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
16 |
|
35847 | 17 |
definition |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
18 |
l_coset :: "[_, 'a, 'a set] \<Rightarrow> 'a set" (infixl "<#\<index>" 60) |
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
19 |
where "a <#\<^bsub>G\<^esub> H = (\<Union>h\<in>H. {a \<otimes>\<^bsub>G\<^esub> h})" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
20 |
|
35847 | 21 |
definition |
14963 | 22 |
RCOSETS :: "[_, 'a set] \<Rightarrow> ('a set)set" ("rcosets\<index> _" [81] 80) |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
23 |
where "rcosets\<^bsub>G\<^esub> H = (\<Union>a\<in>carrier G. {H #>\<^bsub>G\<^esub> a})" |
14963 | 24 |
|
35847 | 25 |
definition |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
26 |
set_mult :: "[_, 'a set ,'a set] \<Rightarrow> 'a set" (infixl "<#>\<index>" 60) |
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
27 |
where "H <#>\<^bsub>G\<^esub> K = (\<Union>h\<in>H. \<Union>k\<in>K. {h \<otimes>\<^bsub>G\<^esub> k})" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
28 |
|
35847 | 29 |
definition |
14963 | 30 |
SET_INV :: "[_,'a set] \<Rightarrow> 'a set" ("set'_inv\<index> _" [81] 80) |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
31 |
where "set_inv\<^bsub>G\<^esub> H = (\<Union>h\<in>H. {inv\<^bsub>G\<^esub> h})" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
32 |
|
14963 | 33 |
|
34 |
locale normal = subgroup + group + |
|
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
35 |
assumes coset_eq: "(\<forall>x \<in> carrier G. H #> x = x <# H)" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
36 |
|
19380 | 37 |
abbreviation |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20318
diff
changeset
|
38 |
normal_rel :: "['a set, ('a, 'b) monoid_scheme] \<Rightarrow> bool" (infixl "\<lhd>" 60) where |
19380 | 39 |
"H \<lhd> G \<equiv> normal H G" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
40 |
|
69749
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
41 |
lemma (in comm_group) subgroup_imp_normal: "subgroup A G \<Longrightarrow> A \<lhd> G" |
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
42 |
by (simp add: normal_def normal_axioms_def is_group l_coset_def r_coset_def m_comm subgroup.mem_carrier) |
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
43 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
44 |
lemma l_coset_eq_set_mult: \<^marker>\<open>contributor \<open>Martin Baillon\<close>\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
45 |
fixes G (structure) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
46 |
shows "x <# H = {x} <#> H" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
47 |
unfolding l_coset_def set_mult_def by simp |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
48 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
49 |
lemma r_coset_eq_set_mult: \<^marker>\<open>contributor \<open>Martin Baillon\<close>\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
50 |
fixes G (structure) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
51 |
shows "H #> x = H <#> {x}" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
52 |
unfolding r_coset_def set_mult_def by simp |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
53 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
54 |
lemma (in subgroup) rcosets_non_empty: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
55 |
assumes "R \<in> rcosets H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
56 |
shows "R \<noteq> {}" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
57 |
proof - |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
58 |
obtain g where "g \<in> carrier G" "R = H #> g" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
59 |
using assms unfolding RCOSETS_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
60 |
hence "\<one> \<otimes> g \<in> R" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
61 |
using one_closed unfolding r_coset_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
62 |
thus ?thesis by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
63 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
64 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
65 |
lemma (in group) diff_neutralizes: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
66 |
assumes "subgroup H G" "R \<in> rcosets H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
67 |
shows "\<And>r1 r2. \<lbrakk> r1 \<in> R; r2 \<in> R \<rbrakk> \<Longrightarrow> r1 \<otimes> (inv r2) \<in> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
68 |
proof - |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
69 |
fix r1 r2 assume r1: "r1 \<in> R" and r2: "r2 \<in> R" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
70 |
obtain g where g: "g \<in> carrier G" "R = H #> g" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
71 |
using assms unfolding RCOSETS_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
72 |
then obtain h1 h2 where h1: "h1 \<in> H" "r1 = h1 \<otimes> g" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
73 |
and h2: "h2 \<in> H" "r2 = h2 \<otimes> g" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
74 |
using r1 r2 unfolding r_coset_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
75 |
hence "r1 \<otimes> (inv r2) = (h1 \<otimes> g) \<otimes> ((inv g) \<otimes> (inv h2))" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
76 |
using inv_mult_group is_group assms(1) g(1) subgroup.mem_carrier by fastforce |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
77 |
also have " ... = (h1 \<otimes> (g \<otimes> inv g) \<otimes> inv h2)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
78 |
using h1 h2 assms(1) g(1) inv_closed m_closed monoid.m_assoc |
68604 | 79 |
monoid_axioms subgroup.mem_carrier |
80 |
proof - |
|
81 |
have "h1 \<in> carrier G" |
|
82 |
by (meson subgroup.mem_carrier assms(1) h1(1)) |
|
83 |
moreover have "h2 \<in> carrier G" |
|
84 |
by (meson subgroup.mem_carrier assms(1) h2(1)) |
|
85 |
ultimately show ?thesis |
|
86 |
using g(1) inv_closed m_assoc m_closed by presburger |
|
87 |
qed |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
88 |
finally have "r1 \<otimes> inv r2 = h1 \<otimes> inv h2" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
89 |
using assms(1) g(1) h1(1) subgroup.mem_carrier by fastforce |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
90 |
thus "r1 \<otimes> inv r2 \<in> H" by (metis assms(1) h1(1) h2(1) subgroup_def) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
91 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
92 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
93 |
lemma mono_set_mult: "\<lbrakk> H \<subseteq> H'; K \<subseteq> K' \<rbrakk> \<Longrightarrow> H <#>\<^bsub>G\<^esub> K \<subseteq> H' <#>\<^bsub>G\<^esub> K'" \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close> |
68517 | 94 |
unfolding set_mult_def by (simp add: UN_mono) |
95 |
||
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
96 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
97 |
subsection \<open>Stable Operations for Subgroups\<close> |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
98 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
99 |
lemma set_mult_consistent [simp]: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close> |
68517 | 100 |
"N <#>\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> K = N <#>\<^bsub>G\<^esub> K" |
101 |
unfolding set_mult_def by simp |
|
102 |
||
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
103 |
lemma r_coset_consistent [simp]: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close> |
68517 | 104 |
"I #>\<^bsub>G \<lparr> carrier := H \<rparr>\<^esub> h = I #>\<^bsub>G\<^esub> h" |
105 |
unfolding r_coset_def by simp |
|
106 |
||
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
107 |
lemma l_coset_consistent [simp]: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close> |
68517 | 108 |
"h <#\<^bsub>G \<lparr> carrier := H \<rparr>\<^esub> I = h <#\<^bsub>G\<^esub> I" |
109 |
unfolding l_coset_def by simp |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
110 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
111 |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
112 |
subsection \<open>Basic Properties of set multiplication\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
113 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
114 |
lemma (in group) setmult_subset_G: |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
115 |
assumes "H \<subseteq> carrier G" "K \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
116 |
shows "H <#> K \<subseteq> carrier G" using assms |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
117 |
by (auto simp add: set_mult_def subsetD) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
118 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
119 |
lemma (in monoid) set_mult_closed: |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
120 |
assumes "H \<subseteq> carrier G" "K \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
121 |
shows "H <#> K \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
122 |
using assms by (auto simp add: set_mult_def subsetD) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
123 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
124 |
lemma (in group) set_mult_assoc: \<^marker>\<open>contributor \<open>Martin Baillon\<close>\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
125 |
assumes "M \<subseteq> carrier G" "H \<subseteq> carrier G" "K \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
126 |
shows "(M <#> H) <#> K = M <#> (H <#> K)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
127 |
proof |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
128 |
show "(M <#> H) <#> K \<subseteq> M <#> (H <#> K)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
129 |
proof |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
130 |
fix x assume "x \<in> (M <#> H) <#> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
131 |
then obtain m h k where x: "m \<in> M" "h \<in> H" "k \<in> K" "x = (m \<otimes> h) \<otimes> k" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
132 |
unfolding set_mult_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
133 |
hence "x = m \<otimes> (h \<otimes> k)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
134 |
using assms m_assoc by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
135 |
thus "x \<in> M <#> (H <#> K)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
136 |
unfolding set_mult_def using x by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
137 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
138 |
next |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
139 |
show "M <#> (H <#> K) \<subseteq> (M <#> H) <#> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
140 |
proof |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
141 |
fix x assume "x \<in> M <#> (H <#> K)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
142 |
then obtain m h k where x: "m \<in> M" "h \<in> H" "k \<in> K" "x = m \<otimes> (h \<otimes> k)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
143 |
unfolding set_mult_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
144 |
hence "x = (m \<otimes> h) \<otimes> k" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
145 |
using assms m_assoc rev_subsetD by metis |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
146 |
thus "x \<in> (M <#> H) <#> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
147 |
unfolding set_mult_def using x by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
148 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
149 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
150 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
151 |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
152 |
|
61382 | 153 |
subsection \<open>Basic Properties of Cosets\<close> |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
154 |
|
14747 | 155 |
lemma (in group) coset_mult_assoc: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
156 |
assumes "M \<subseteq> carrier G" "g \<in> carrier G" "h \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
157 |
shows "(M #> g) #> h = M #> (g \<otimes> h)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
158 |
using assms by (force simp add: r_coset_def m_assoc) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
159 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
160 |
lemma (in group) coset_assoc: |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
161 |
assumes "x \<in> carrier G" "y \<in> carrier G" "H \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
162 |
shows "x <# (H #> y) = (x <# H) #> y" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
163 |
using set_mult_assoc[of "{x}" H "{y}"] |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
164 |
by (simp add: l_coset_eq_set_mult r_coset_eq_set_mult assms) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
165 |
|
14747 | 166 |
lemma (in group) coset_mult_one [simp]: "M \<subseteq> carrier G ==> M #> \<one> = M" |
167 |
by (force simp add: r_coset_def) |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
168 |
|
14747 | 169 |
lemma (in group) coset_mult_inv1: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
170 |
assumes "M #> (x \<otimes> (inv y)) = M" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
171 |
and "x \<in> carrier G" "y \<in> carrier G" "M \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
172 |
shows "M #> x = M #> y" using assms |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
173 |
by (metis coset_mult_assoc group.inv_solve_right is_group subgroup_def subgroup_self) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
174 |
|
14747 | 175 |
lemma (in group) coset_mult_inv2: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
176 |
assumes "M #> x = M #> y" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
177 |
and "x \<in> carrier G" "y \<in> carrier G" "M \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
178 |
shows "M #> (x \<otimes> (inv y)) = M " using assms |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
179 |
by (metis group.coset_mult_assoc group.coset_mult_one inv_closed is_group r_inv) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
180 |
|
14747 | 181 |
lemma (in group) coset_join1: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
182 |
assumes "H #> x = H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
183 |
and "x \<in> carrier G" "subgroup H G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
184 |
shows "x \<in> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
185 |
using assms r_coset_def l_one subgroup.one_closed sym by fastforce |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
186 |
|
14747 | 187 |
lemma (in group) solve_equation: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
188 |
assumes "subgroup H G" "x \<in> H" "y \<in> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
189 |
shows "\<exists>h \<in> H. y = h \<otimes> x" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
190 |
proof - |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
191 |
have "y = (y \<otimes> (inv x)) \<otimes> x" using assms |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
192 |
by (simp add: m_assoc subgroup.mem_carrier) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
193 |
moreover have "y \<otimes> (inv x) \<in> H" using assms |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
194 |
by (simp add: subgroup_def) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
195 |
ultimately show ?thesis by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
196 |
qed |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
197 |
|
69749
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
198 |
lemma (in group_hom) inj_on_one_iff: |
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
199 |
"inj_on h (carrier G) \<longleftrightarrow> (\<forall>x. x \<in> carrier G \<longrightarrow> h x = one H \<longrightarrow> x = one G)" |
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
200 |
using G.solve_equation G.subgroup_self by (force simp: inj_on_def) |
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
201 |
|
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
202 |
lemma inj_on_one_iff': |
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
203 |
"\<lbrakk>h \<in> hom G H; group G; group H\<rbrakk> \<Longrightarrow> inj_on h (carrier G) \<longleftrightarrow> (\<forall>x. x \<in> carrier G \<longrightarrow> h x = one H \<longrightarrow> x = one G)" |
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
204 |
using group_hom.inj_on_one_iff group_hom.intro group_hom_axioms.intro by blast |
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
205 |
|
70027
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
206 |
lemma mon_iff_hom_one: |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
207 |
"\<lbrakk>group G; group H\<rbrakk> \<Longrightarrow> f \<in> mon G H \<longleftrightarrow> f \<in> hom G H \<and> (\<forall>x. x \<in> carrier G \<and> f x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>)" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
208 |
by (auto simp: mon_def inj_on_one_iff') |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
209 |
|
69749
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
210 |
lemma (in group_hom) iso_iff: "h \<in> iso G H \<longleftrightarrow> carrier H \<subseteq> h ` carrier G \<and> (\<forall>x\<in>carrier G. h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>)" |
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
211 |
by (auto simp: iso_def bij_betw_def inj_on_one_iff) |
10e48c47a549
some new results in group theory
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
212 |
|
14963 | 213 |
lemma (in group) repr_independence: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
214 |
assumes "y \<in> H #> x" "x \<in> carrier G" "subgroup H G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
215 |
shows "H #> x = H #> y" using assms |
14963 | 216 |
by (auto simp add: r_coset_def m_assoc [symmetric] |
217 |
subgroup.subset [THEN subsetD] |
|
218 |
subgroup.m_closed solve_equation) |
|
219 |
||
14747 | 220 |
lemma (in group) coset_join2: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
221 |
assumes "x \<in> carrier G" "subgroup H G" "x \<in> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
222 |
shows "H #> x = H" using assms |
69597 | 223 |
\<comment> \<open>Alternative proof is to put \<^term>\<open>x=\<one>\<close> in \<open>repr_independence\<close>.\<close> |
14963 | 224 |
by (force simp add: subgroup.m_closed r_coset_def solve_equation) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
225 |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
226 |
lemma (in group) coset_join3: |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
227 |
assumes "x \<in> carrier G" "subgroup H G" "x \<in> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
228 |
shows "x <# H = H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
229 |
proof |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
230 |
have "\<And>h. h \<in> H \<Longrightarrow> x \<otimes> h \<in> H" using assms |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
231 |
by (simp add: subgroup.m_closed) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
232 |
thus "x <# H \<subseteq> H" unfolding l_coset_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
233 |
next |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
234 |
have "\<And>h. h \<in> H \<Longrightarrow> x \<otimes> ((inv x) \<otimes> h) = h" |
68604 | 235 |
by (metis (no_types, lifting) assms group.inv_closed group.inv_solve_left is_group |
236 |
monoid.m_closed monoid_axioms subgroup.mem_carrier) |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
237 |
moreover have "\<And>h. h \<in> H \<Longrightarrow> (inv x) \<otimes> h \<in> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
238 |
by (simp add: assms subgroup.m_closed subgroup.m_inv_closed) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
239 |
ultimately show "H \<subseteq> x <# H" unfolding l_coset_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
240 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
241 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
242 |
lemma (in monoid) r_coset_subset_G: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
243 |
"\<lbrakk> H \<subseteq> carrier G; x \<in> carrier G \<rbrakk> \<Longrightarrow> H #> x \<subseteq> carrier G" |
14747 | 244 |
by (auto simp add: r_coset_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
245 |
|
14747 | 246 |
lemma (in group) rcosI: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
247 |
"\<lbrakk> h \<in> H; H \<subseteq> carrier G; x \<in> carrier G \<rbrakk> \<Longrightarrow> h \<otimes> x \<in> H #> x" |
14747 | 248 |
by (auto simp add: r_coset_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
249 |
|
14963 | 250 |
lemma (in group) rcosetsI: |
251 |
"\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H #> x \<in> rcosets H" |
|
252 |
by (auto simp add: RCOSETS_def) |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
253 |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
254 |
lemma (in group) rcos_self: |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
255 |
"\<lbrakk> x \<in> carrier G; subgroup H G \<rbrakk> \<Longrightarrow> x \<in> H #> x" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
256 |
by (metis l_one rcosI subgroup_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
257 |
|
61382 | 258 |
text (in group) \<open>Opposite of @{thm [source] "repr_independence"}\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
259 |
lemma (in group) repr_independenceD: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
260 |
assumes "subgroup H G" "y \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
261 |
and "H #> x = H #> y" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
262 |
shows "y \<in> H #> x" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
263 |
using assms by (simp add: rcos_self) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
264 |
|
61382 | 265 |
text \<open>Elements of a right coset are in the carrier\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
266 |
lemma (in subgroup) elemrcos_carrier: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
267 |
assumes "group G" "a \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
268 |
and "a' \<in> H #> a" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
269 |
shows "a' \<in> carrier G" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
270 |
by (meson assms group.is_monoid monoid.r_coset_subset_G subset subsetCE) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
271 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
272 |
lemma (in subgroup) rcos_const: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
273 |
assumes "group G" "h \<in> H" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
274 |
shows "H #> h = H" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
275 |
using group.coset_join2[OF assms(1), of h H] |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
276 |
by (simp add: assms(2) subgroup_axioms) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
277 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
278 |
lemma (in subgroup) rcos_module_imp: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
279 |
assumes "group G" "x \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
280 |
and "x' \<in> H #> x" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
281 |
shows "(x' \<otimes> inv x) \<in> H" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
282 |
proof - |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
283 |
obtain h where h: "h \<in> H" "x' = h \<otimes> x" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
284 |
using assms(3) unfolding r_coset_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
285 |
hence "x' \<otimes> inv x = h" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
286 |
by (metis assms elemrcos_carrier group.inv_solve_right mem_carrier) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
287 |
thus ?thesis using h by blast |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
288 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
289 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
290 |
lemma (in subgroup) rcos_module_rev: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
291 |
assumes "group G" "x \<in> carrier G" "x' \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
292 |
and "(x' \<otimes> inv x) \<in> H" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
293 |
shows "x' \<in> H #> x" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
294 |
proof - |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
295 |
obtain h where h: "h \<in> H" "x' \<otimes> inv x = h" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
296 |
using assms(4) unfolding r_coset_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
297 |
hence "x' = h \<otimes> x" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
298 |
by (metis assms group.inv_solve_right mem_carrier) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
299 |
thus ?thesis using h unfolding r_coset_def by blast |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
300 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
301 |
|
61382 | 302 |
text \<open>Module property of right cosets\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
303 |
lemma (in subgroup) rcos_module: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
304 |
assumes "group G" "x \<in> carrier G" "x' \<in> carrier G" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
305 |
shows "(x' \<in> H #> x) = (x' \<otimes> inv x \<in> H)" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
306 |
using rcos_module_rev rcos_module_imp assms by blast |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
307 |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
308 |
text \<open>Right cosets are subsets of the carrier.\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
309 |
lemma (in subgroup) rcosets_carrier: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
310 |
assumes "group G" "X \<in> rcosets H" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
311 |
shows "X \<subseteq> carrier G" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
312 |
using assms elemrcos_carrier singletonD |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
313 |
subset_eq unfolding RCOSETS_def by force |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
314 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
315 |
|
61382 | 316 |
text \<open>Multiplication of general subsets\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
317 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
318 |
lemma (in comm_group) mult_subgroups: |
68604 | 319 |
assumes HG: "subgroup H G" and KG: "subgroup K G" |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
320 |
shows "subgroup (H <#> K) G" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
321 |
proof (rule subgroup.intro) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
322 |
show "H <#> K \<subseteq> carrier G" |
68452
c027dfbfad30
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
paulson <lp15@cam.ac.uk>
parents:
68445
diff
changeset
|
323 |
by (simp add: setmult_subset_G assms subgroup.subset) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
324 |
next |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
325 |
have "\<one> \<otimes> \<one> \<in> H <#> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
326 |
unfolding set_mult_def using assms subgroup.one_closed by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
327 |
thus "\<one> \<in> H <#> K" by simp |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
328 |
next |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
329 |
show "\<And>x. x \<in> H <#> K \<Longrightarrow> inv x \<in> H <#> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
330 |
proof - |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
331 |
fix x assume "x \<in> H <#> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
332 |
then obtain h k where hk: "h \<in> H" "k \<in> K" "x = h \<otimes> k" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
333 |
unfolding set_mult_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
334 |
hence "inv x = (inv k) \<otimes> (inv h)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
335 |
by (meson inv_mult_group assms subgroup.mem_carrier) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
336 |
hence "inv x = (inv h) \<otimes> (inv k)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
337 |
by (metis hk inv_mult assms subgroup.mem_carrier) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
338 |
thus "inv x \<in> H <#> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
339 |
unfolding set_mult_def using hk assms |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
340 |
by (metis (no_types, lifting) UN_iff singletonI subgroup_def) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
341 |
qed |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
342 |
next |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
343 |
show "\<And>x y. x \<in> H <#> K \<Longrightarrow> y \<in> H <#> K \<Longrightarrow> x \<otimes> y \<in> H <#> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
344 |
proof - |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
345 |
fix x y assume "x \<in> H <#> K" "y \<in> H <#> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
346 |
then obtain h1 k1 h2 k2 where h1k1: "h1 \<in> H" "k1 \<in> K" "x = h1 \<otimes> k1" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
347 |
and h2k2: "h2 \<in> H" "k2 \<in> K" "y = h2 \<otimes> k2" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
348 |
unfolding set_mult_def by blast |
68604 | 349 |
with KG HG have carr: "k1 \<in> carrier G" "h1 \<in> carrier G" "k2 \<in> carrier G" "h2 \<in> carrier G" |
350 |
by (meson subgroup.mem_carrier)+ |
|
351 |
have "x \<otimes> y = (h1 \<otimes> k1) \<otimes> (h2 \<otimes> k2)" |
|
352 |
using h1k1 h2k2 by simp |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
353 |
also have " ... = h1 \<otimes> (k1 \<otimes> h2) \<otimes> k2" |
68604 | 354 |
by (simp add: carr comm_groupE(3) comm_group_axioms) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
355 |
also have " ... = h1 \<otimes> (h2 \<otimes> k1) \<otimes> k2" |
68604 | 356 |
by (simp add: carr m_comm) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
357 |
finally have "x \<otimes> y = (h1 \<otimes> h2) \<otimes> (k1 \<otimes> k2)" |
68604 | 358 |
by (simp add: carr comm_groupE(3) comm_group_axioms) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
359 |
thus "x \<otimes> y \<in> H <#> K" unfolding set_mult_def |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
360 |
using subgroup.m_closed[OF assms(1) h1k1(1) h2k2(1)] |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
361 |
subgroup.m_closed[OF assms(2) h1k1(2) h2k2(2)] by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
362 |
qed |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
363 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
364 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
365 |
lemma (in subgroup) lcos_module_rev: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
366 |
assumes "group G" "x \<in> carrier G" "x' \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
367 |
and "(inv x \<otimes> x') \<in> H" |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
368 |
shows "x' \<in> x <# H" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
369 |
proof - |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
370 |
obtain h where h: "h \<in> H" "inv x \<otimes> x' = h" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
371 |
using assms(4) unfolding l_coset_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
372 |
hence "x' = x \<otimes> h" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
373 |
by (metis assms group.inv_solve_left mem_carrier) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
374 |
thus ?thesis using h unfolding l_coset_def by blast |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
375 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
376 |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
377 |
|
61382 | 378 |
subsection \<open>Normal subgroups\<close> |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
379 |
|
14963 | 380 |
lemma normal_imp_subgroup: "H \<lhd> G \<Longrightarrow> subgroup H G" |
70027
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
381 |
by (rule normal.axioms(1)) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
382 |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
383 |
lemma (in group) normalI: |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
384 |
"subgroup H G \<Longrightarrow> (\<forall>x \<in> carrier G. H #> x = x <# H) \<Longrightarrow> H \<lhd> G" |
41528 | 385 |
by (simp add: normal_def normal_axioms_def is_group) |
14963 | 386 |
|
387 |
lemma (in normal) inv_op_closed1: |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
388 |
assumes "x \<in> carrier G" and "h \<in> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
389 |
shows "(inv x) \<otimes> h \<otimes> x \<in> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
390 |
proof - |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
391 |
have "h \<otimes> x \<in> x <# H" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
392 |
using assms coset_eq assms(1) unfolding r_coset_def by blast |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
393 |
then obtain h' where "h' \<in> H" "h \<otimes> x = x \<otimes> h'" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
394 |
unfolding l_coset_def by blast |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
395 |
thus ?thesis by (metis assms inv_closed l_inv l_one m_assoc mem_carrier) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
396 |
qed |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
397 |
|
14963 | 398 |
lemma (in normal) inv_op_closed2: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
399 |
assumes "x \<in> carrier G" and "h \<in> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
400 |
shows "x \<otimes> h \<otimes> (inv x) \<in> H" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
401 |
using assms inv_op_closed1 by (metis inv_closed inv_inv) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
402 |
|
70027
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
403 |
lemma (in comm_group) normal_iff_subgroup: |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
404 |
"N \<lhd> G \<longleftrightarrow> subgroup N G" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
405 |
proof |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
406 |
assume "subgroup N G" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
407 |
then show "N \<lhd> G" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
408 |
by unfold_locales (auto simp: subgroupE subgroup.one_closed l_coset_def r_coset_def m_comm subgroup.mem_carrier) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
409 |
qed (simp add: normal_imp_subgroup) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
410 |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
411 |
|
61382 | 412 |
text\<open>Alternative characterization of normal subgroups\<close> |
14747 | 413 |
lemma (in group) normal_inv_iff: |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
414 |
"(N \<lhd> G) = |
67091 | 415 |
(subgroup N G \<and> (\<forall>x \<in> carrier G. \<forall>h \<in> N. x \<otimes> h \<otimes> (inv x) \<in> N))" |
14747 | 416 |
(is "_ = ?rhs") |
417 |
proof |
|
418 |
assume N: "N \<lhd> G" |
|
419 |
show ?rhs |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
420 |
by (blast intro: N normal.inv_op_closed2 normal_imp_subgroup) |
14747 | 421 |
next |
422 |
assume ?rhs |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
423 |
hence sg: "subgroup N G" |
14963 | 424 |
and closed: "\<And>x. x\<in>carrier G \<Longrightarrow> \<forall>h\<in>N. x \<otimes> h \<otimes> inv x \<in> N" by auto |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
425 |
hence sb: "N \<subseteq> carrier G" by (simp add: subgroup.subset) |
14747 | 426 |
show "N \<lhd> G" |
14963 | 427 |
proof (intro normalI [OF sg], simp add: l_coset_def r_coset_def, clarify) |
14747 | 428 |
fix x |
429 |
assume x: "x \<in> carrier G" |
|
15120 | 430 |
show "(\<Union>h\<in>N. {h \<otimes> x}) = (\<Union>h\<in>N. {x \<otimes> h})" |
14747 | 431 |
proof |
15120 | 432 |
show "(\<Union>h\<in>N. {h \<otimes> x}) \<subseteq> (\<Union>h\<in>N. {x \<otimes> h})" |
14747 | 433 |
proof clarify |
434 |
fix n |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
435 |
assume n: "n \<in> N" |
15120 | 436 |
show "n \<otimes> x \<in> (\<Union>h\<in>N. {x \<otimes> h})" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
437 |
proof |
14963 | 438 |
from closed [of "inv x"] |
439 |
show "inv x \<otimes> n \<otimes> x \<in> N" by (simp add: x n) |
|
440 |
show "n \<otimes> x \<in> {x \<otimes> (inv x \<otimes> n \<otimes> x)}" |
|
14747 | 441 |
by (simp add: x n m_assoc [symmetric] sb [THEN subsetD]) |
442 |
qed |
|
443 |
qed |
|
444 |
next |
|
15120 | 445 |
show "(\<Union>h\<in>N. {x \<otimes> h}) \<subseteq> (\<Union>h\<in>N. {h \<otimes> x})" |
14747 | 446 |
proof clarify |
447 |
fix n |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
448 |
assume n: "n \<in> N" |
15120 | 449 |
show "x \<otimes> n \<in> (\<Union>h\<in>N. {h \<otimes> x})" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
450 |
proof |
14963 | 451 |
show "x \<otimes> n \<otimes> inv x \<in> N" by (simp add: x n closed) |
452 |
show "x \<otimes> n \<in> {x \<otimes> n \<otimes> inv x \<otimes> x}" |
|
14747 | 453 |
by (simp add: x n m_assoc sb [THEN subsetD]) |
454 |
qed |
|
455 |
qed |
|
456 |
qed |
|
457 |
qed |
|
458 |
qed |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
459 |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
460 |
corollary (in group) normal_invI: |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
461 |
assumes "subgroup N G" and "\<And>x h. \<lbrakk> x \<in> carrier G; h \<in> N \<rbrakk> \<Longrightarrow> x \<otimes> h \<otimes> inv x \<in> N" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
462 |
shows "N \<lhd> G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
463 |
using assms normal_inv_iff by blast |
14963 | 464 |
|
69122
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
465 |
corollary (in group) normal_invE: |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
466 |
assumes "N \<lhd> G" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
467 |
shows "subgroup N G" and "\<And>x h. \<lbrakk> x \<in> carrier G; h \<in> N \<rbrakk> \<Longrightarrow> x \<otimes> h \<otimes> inv x \<in> N" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
468 |
using assms normal_inv_iff apply blast |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
469 |
by (simp add: assms normal.inv_op_closed2) |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
470 |
|
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
471 |
|
68687 | 472 |
lemma (in group) one_is_normal: "{\<one>} \<lhd> G" |
473 |
proof(intro normal_invI) |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
474 |
show "subgroup {\<one>} G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
475 |
by (simp add: subgroup_def) |
68687 | 476 |
qed simp |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
477 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
478 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
479 |
subsection\<open>More Properties of Left Cosets\<close> |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
480 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
481 |
lemma (in group) l_repr_independence: |
68687 | 482 |
assumes "y \<in> x <# H" "x \<in> carrier G" and HG: "subgroup H G" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
483 |
shows "x <# H = y <# H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
484 |
proof - |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
485 |
obtain h' where h': "h' \<in> H" "y = x \<otimes> h'" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
486 |
using assms(1) unfolding l_coset_def by blast |
68604 | 487 |
hence "x \<otimes> h = y \<otimes> ((inv h') \<otimes> h)" if "h \<in> H" for h |
488 |
proof - |
|
68687 | 489 |
have "h' \<in> carrier G" |
490 |
by (meson HG h'(1) subgroup.mem_carrier) |
|
491 |
moreover have "h \<in> carrier G" |
|
492 |
by (meson HG subgroup.mem_carrier that) |
|
493 |
ultimately show ?thesis |
|
494 |
by (metis assms(2) h'(2) inv_closed inv_solve_right m_assoc m_closed) |
|
68604 | 495 |
qed |
68687 | 496 |
hence "\<And>xh. xh \<in> x <# H \<Longrightarrow> xh \<in> y <# H" |
497 |
unfolding l_coset_def by (metis (no_types, lifting) UN_iff HG h'(1) subgroup_def) |
|
498 |
moreover have "\<And>h. h \<in> H \<Longrightarrow> y \<otimes> h = x \<otimes> (h' \<otimes> h)" |
|
499 |
using h' by (meson assms(2) HG m_assoc subgroup.mem_carrier) |
|
500 |
hence "\<And>yh. yh \<in> y <# H \<Longrightarrow> yh \<in> x <# H" |
|
501 |
unfolding l_coset_def using subgroup.m_closed[OF HG h'(1)] by blast |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
502 |
ultimately show ?thesis by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
503 |
qed |
14803 | 504 |
|
14747 | 505 |
lemma (in group) lcos_m_assoc: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
506 |
"\<lbrakk> M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G \<rbrakk> \<Longrightarrow> g <# (h <# M) = (g \<otimes> h) <# M" |
14747 | 507 |
by (force simp add: l_coset_def m_assoc) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
508 |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
509 |
lemma (in group) lcos_mult_one: "M \<subseteq> carrier G \<Longrightarrow> \<one> <# M = M" |
14747 | 510 |
by (force simp add: l_coset_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
511 |
|
14747 | 512 |
lemma (in group) l_coset_subset_G: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
513 |
"\<lbrakk> H \<subseteq> carrier G; x \<in> carrier G \<rbrakk> \<Longrightarrow> x <# H \<subseteq> carrier G" |
14747 | 514 |
by (auto simp add: l_coset_def subsetD) |
515 |
||
516 |
lemma (in group) l_coset_carrier: |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
517 |
"\<lbrakk> y \<in> x <# H; x \<in> carrier G; subgroup H G \<rbrakk> \<Longrightarrow> y \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
518 |
by (auto simp add: l_coset_def m_assoc subgroup.subset [THEN subsetD] subgroup.m_closed) |
14530 | 519 |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
520 |
lemma (in group) l_coset_swap: |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
521 |
assumes "y \<in> x <# H" "x \<in> carrier G" "subgroup H G" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
522 |
shows "x \<in> y <# H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
523 |
using assms(2) l_repr_independence[OF assms] subgroup.one_closed[OF assms(3)] |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
524 |
unfolding l_coset_def by fastforce |
14530 | 525 |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
526 |
lemma (in group) subgroup_mult_id: |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
527 |
assumes "subgroup H G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
528 |
shows "H <#> H = H" |
14666 | 529 |
proof |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
530 |
show "H <#> H \<subseteq> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
531 |
unfolding set_mult_def using subgroup.m_closed[OF assms] by (simp add: UN_subset_iff) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
532 |
show "H \<subseteq> H <#> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
533 |
proof |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
534 |
fix x assume x: "x \<in> H" thus "x \<in> H <#> H" unfolding set_mult_def |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
535 |
using subgroup.m_closed[OF assms subgroup.one_closed[OF assms] x] subgroup.one_closed[OF assms] |
68604 | 536 |
using assms subgroup.mem_carrier by force |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
537 |
qed |
14530 | 538 |
qed |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
539 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
540 |
|
63167 | 541 |
subsubsection \<open>Set of Inverses of an \<open>r_coset\<close>.\<close> |
14666 | 542 |
|
14963 | 543 |
lemma (in normal) rcos_inv: |
544 |
assumes x: "x \<in> carrier G" |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
545 |
shows "set_inv (H #> x) = H #> (inv x)" |
14963 | 546 |
proof (simp add: r_coset_def SET_INV_def x inv_mult_group, safe) |
547 |
fix h |
|
41528 | 548 |
assume h: "h \<in> H" |
15120 | 549 |
show "inv x \<otimes> inv h \<in> (\<Union>j\<in>H. {j \<otimes> inv x})" |
14963 | 550 |
proof |
551 |
show "inv x \<otimes> inv h \<otimes> x \<in> H" |
|
41528 | 552 |
by (simp add: inv_op_closed1 h x) |
14963 | 553 |
show "inv x \<otimes> inv h \<in> {inv x \<otimes> inv h \<otimes> x \<otimes> inv x}" |
41528 | 554 |
by (simp add: h x m_assoc) |
14963 | 555 |
qed |
15120 | 556 |
show "h \<otimes> inv x \<in> (\<Union>j\<in>H. {inv x \<otimes> inv j})" |
14963 | 557 |
proof |
558 |
show "x \<otimes> inv h \<otimes> inv x \<in> H" |
|
41528 | 559 |
by (simp add: inv_op_closed2 h x) |
14963 | 560 |
show "h \<otimes> inv x \<in> {inv x \<otimes> inv (x \<otimes> inv h \<otimes> inv x)}" |
41528 | 561 |
by (simp add: h x m_assoc [symmetric] inv_mult_group) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
562 |
qed |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
563 |
qed |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
564 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
565 |
|
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
566 |
subsubsection \<open>Theorems for \<open><#>\<close> with \<open>#>\<close> or \<open><#\<close>.\<close> |
14666 | 567 |
|
14747 | 568 |
lemma (in group) setmult_rcos_assoc: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
569 |
"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
570 |
H <#> (K #> x) = (H <#> K) #> x" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
571 |
using set_mult_assoc[of H K "{x}"] by (simp add: r_coset_eq_set_mult) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
572 |
|
14747 | 573 |
lemma (in group) rcos_assoc_lcos: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
574 |
"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
575 |
(H #> x) <#> K = H <#> (x <# K)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
576 |
using set_mult_assoc[of H "{x}" K] |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
577 |
by (simp add: l_coset_eq_set_mult r_coset_eq_set_mult) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
578 |
|
14963 | 579 |
lemma (in normal) rcos_mult_step1: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
580 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
581 |
(H #> x) <#> (H #> y) = (H <#> (x <# H)) #> y" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
582 |
by (simp add: setmult_rcos_assoc r_coset_subset_G |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
583 |
subset l_coset_subset_G rcos_assoc_lcos) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
584 |
|
14963 | 585 |
lemma (in normal) rcos_mult_step2: |
586 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> |
|
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
587 |
\<Longrightarrow> (H <#> (x <# H)) #> y = (H <#> (H #> x)) #> y" |
14963 | 588 |
by (insert coset_eq, simp add: normal_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
589 |
|
14963 | 590 |
lemma (in normal) rcos_mult_step3: |
591 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> |
|
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
592 |
\<Longrightarrow> (H <#> (H #> x)) #> y = H #> (x \<otimes> y)" |
14963 | 593 |
by (simp add: setmult_rcos_assoc coset_mult_assoc |
41528 | 594 |
subgroup_mult_id normal.axioms subset normal_axioms) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
595 |
|
14963 | 596 |
lemma (in normal) rcos_sum: |
597 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> |
|
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
598 |
\<Longrightarrow> (H #> x) <#> (H #> y) = H #> (x \<otimes> y)" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
599 |
by (simp add: rcos_mult_step1 rcos_mult_step2 rcos_mult_step3) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
600 |
|
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
601 |
lemma (in normal) rcosets_mult_eq: "M \<in> rcosets H \<Longrightarrow> H <#> M = M" |
63167 | 602 |
\<comment> \<open>generalizes \<open>subgroup_mult_id\<close>\<close> |
14963 | 603 |
by (auto simp add: RCOSETS_def subset |
41528 | 604 |
setmult_rcos_assoc subgroup_mult_id normal.axioms normal_axioms) |
14963 | 605 |
|
606 |
||
61382 | 607 |
subsubsection\<open>An Equivalence Relation\<close> |
14963 | 608 |
|
35847 | 609 |
definition |
610 |
r_congruent :: "[('a,'b)monoid_scheme, 'a set] \<Rightarrow> ('a*'a)set" ("rcong\<index> _") |
|
67091 | 611 |
where "rcong\<^bsub>G\<^esub> H = {(x,y). x \<in> carrier G \<and> y \<in> carrier G \<and> inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> y \<in> H}" |
14963 | 612 |
|
613 |
||
614 |
lemma (in subgroup) equiv_rcong: |
|
27611 | 615 |
assumes "group G" |
14963 | 616 |
shows "equiv (carrier G) (rcong H)" |
27611 | 617 |
proof - |
29237 | 618 |
interpret group G by fact |
27611 | 619 |
show ?thesis |
40815 | 620 |
proof (intro equivI) |
30198 | 621 |
show "refl_on (carrier G) (rcong H)" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
622 |
by (auto simp add: r_congruent_def refl_on_def) |
27611 | 623 |
next |
624 |
show "sym (rcong H)" |
|
625 |
proof (simp add: r_congruent_def sym_def, clarify) |
|
626 |
fix x y |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
627 |
assume [simp]: "x \<in> carrier G" "y \<in> carrier G" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31727
diff
changeset
|
628 |
and "inv x \<otimes> y \<in> H" |
46721 | 629 |
hence "inv (inv x \<otimes> y) \<in> H" by simp |
27611 | 630 |
thus "inv y \<otimes> x \<in> H" by (simp add: inv_mult_group) |
631 |
qed |
|
632 |
next |
|
633 |
show "trans (rcong H)" |
|
634 |
proof (simp add: r_congruent_def trans_def, clarify) |
|
635 |
fix x y z |
|
636 |
assume [simp]: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31727
diff
changeset
|
637 |
and "inv x \<otimes> y \<in> H" and "inv y \<otimes> z \<in> H" |
27611 | 638 |
hence "(inv x \<otimes> y) \<otimes> (inv y \<otimes> z) \<in> H" by simp |
27698 | 639 |
hence "inv x \<otimes> (y \<otimes> inv y) \<otimes> z \<in> H" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
640 |
by (simp add: m_assoc del: r_inv Units_r_inv) |
27611 | 641 |
thus "inv x \<otimes> z \<in> H" by simp |
642 |
qed |
|
14963 | 643 |
qed |
644 |
qed |
|
645 |
||
63167 | 646 |
text\<open>Equivalence classes of \<open>rcong\<close> correspond to left cosets. |
14963 | 647 |
Was there a mistake in the definitions? I'd have expected them to |
61382 | 648 |
correspond to right cosets.\<close> |
14963 | 649 |
|
650 |
(* CB: This is correct, but subtle. |
|
651 |
We call H #> a the right coset of a relative to H. According to |
|
652 |
Jacobson, this is what the majority of group theory literature does. |
|
653 |
He then defines the notion of congruence relation ~ over monoids as |
|
654 |
equivalence relation with a ~ a' & b ~ b' \<Longrightarrow> a*b ~ a'*b'. |
|
655 |
Our notion of right congruence induced by K: rcong K appears only in |
|
656 |
the context where K is a normal subgroup. Jacobson doesn't name it. |
|
657 |
But in this context left and right cosets are identical. |
|
658 |
*) |
|
659 |
||
660 |
lemma (in subgroup) l_coset_eq_rcong: |
|
27611 | 661 |
assumes "group G" |
14963 | 662 |
assumes a: "a \<in> carrier G" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
663 |
shows "a <# H = (rcong H) `` {a}" |
27611 | 664 |
proof - |
29237 | 665 |
interpret group G by fact |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
666 |
show ?thesis by (force simp add: r_congruent_def l_coset_def m_assoc [symmetric] a ) |
27611 | 667 |
qed |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
668 |
|
35849 | 669 |
|
61382 | 670 |
subsubsection\<open>Two Distinct Right Cosets are Disjoint\<close> |
14803 | 671 |
|
672 |
lemma (in group) rcos_equation: |
|
27611 | 673 |
assumes "subgroup H G" |
674 |
assumes p: "ha \<otimes> a = h \<otimes> b" "a \<in> carrier G" "b \<in> carrier G" "h \<in> H" "ha \<in> H" "hb \<in> H" |
|
675 |
shows "hb \<otimes> a \<in> (\<Union>h\<in>H. {h \<otimes> b})" |
|
676 |
proof - |
|
29237 | 677 |
interpret subgroup H G by fact |
68687 | 678 |
from p show ?thesis |
679 |
by (rule_tac UN_I [of "hb \<otimes> ((inv ha) \<otimes> h)"]) (auto simp: inv_solve_left m_assoc) |
|
27611 | 680 |
qed |
14803 | 681 |
|
682 |
lemma (in group) rcos_disjoint: |
|
27611 | 683 |
assumes "subgroup H G" |
68975
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68687
diff
changeset
|
684 |
shows "pairwise disjnt (rcosets H)" |
27611 | 685 |
proof - |
29237 | 686 |
interpret subgroup H G by fact |
68975
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68687
diff
changeset
|
687 |
show ?thesis |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68687
diff
changeset
|
688 |
unfolding RCOSETS_def r_coset_def pairwise_def disjnt_def |
68687 | 689 |
by (blast intro: rcos_equation assms sym) |
27611 | 690 |
qed |
14803 | 691 |
|
35849 | 692 |
|
63167 | 693 |
subsection \<open>Further lemmas for \<open>r_congruent\<close>\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
694 |
|
61382 | 695 |
text \<open>The relation is a congruence\<close> |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
696 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
697 |
lemma (in normal) congruent_rcong: |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
698 |
shows "congruent2 (rcong H) (rcong H) (\<lambda>a b. a \<otimes> b <# H)" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
699 |
proof (intro congruent2I[of "carrier G" _ "carrier G" _] equiv_rcong is_group) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
700 |
fix a b c |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
701 |
assume abrcong: "(a, b) \<in> rcong H" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
702 |
and ccarr: "c \<in> carrier G" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
703 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
704 |
from abrcong |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
705 |
have acarr: "a \<in> carrier G" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
706 |
and bcarr: "b \<in> carrier G" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
707 |
and abH: "inv a \<otimes> b \<in> H" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
708 |
unfolding r_congruent_def |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
709 |
by fast+ |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
710 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
711 |
note carr = acarr bcarr ccarr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
712 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
713 |
from ccarr and abH |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
714 |
have "inv c \<otimes> (inv a \<otimes> b) \<otimes> c \<in> H" by (rule inv_op_closed1) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
715 |
moreover |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
716 |
from carr and inv_closed |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
717 |
have "inv c \<otimes> (inv a \<otimes> b) \<otimes> c = (inv c \<otimes> inv a) \<otimes> (b \<otimes> c)" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
718 |
by (force cong: m_assoc) |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
719 |
moreover |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
720 |
from carr and inv_closed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
721 |
have "\<dots> = (inv (a \<otimes> c)) \<otimes> (b \<otimes> c)" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
722 |
by (simp add: inv_mult_group) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
723 |
ultimately |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
724 |
have "(inv (a \<otimes> c)) \<otimes> (b \<otimes> c) \<in> H" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
725 |
from carr and this |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
726 |
have "(b \<otimes> c) \<in> (a \<otimes> c) <# H" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
727 |
by (simp add: lcos_module_rev[OF is_group]) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
728 |
from carr and this and is_subgroup |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
729 |
show "(a \<otimes> c) <# H = (b \<otimes> c) <# H" by (intro l_repr_independence, simp+) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
730 |
next |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
731 |
fix a b c |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
732 |
assume abrcong: "(a, b) \<in> rcong H" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
733 |
and ccarr: "c \<in> carrier G" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
734 |
|
46721 | 735 |
from ccarr have "c \<in> Units G" by simp |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
736 |
hence cinvc_one: "inv c \<otimes> c = \<one>" by (rule Units_l_inv) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
737 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
738 |
from abrcong |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
739 |
have acarr: "a \<in> carrier G" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
740 |
and bcarr: "b \<in> carrier G" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
741 |
and abH: "inv a \<otimes> b \<in> H" |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
742 |
by (unfold r_congruent_def, fast+) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
743 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
744 |
note carr = acarr bcarr ccarr |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
745 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
746 |
from carr and inv_closed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
747 |
have "inv a \<otimes> b = inv a \<otimes> (\<one> \<otimes> b)" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
748 |
also from carr and inv_closed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
749 |
have "\<dots> = inv a \<otimes> (inv c \<otimes> c) \<otimes> b" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
750 |
also from carr and inv_closed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
751 |
have "\<dots> = (inv a \<otimes> inv c) \<otimes> (c \<otimes> b)" by (force cong: m_assoc) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
752 |
also from carr and inv_closed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
753 |
have "\<dots> = inv (c \<otimes> a) \<otimes> (c \<otimes> b)" by (simp add: inv_mult_group) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
754 |
finally |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
755 |
have "inv a \<otimes> b = inv (c \<otimes> a) \<otimes> (c \<otimes> b)" . |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
756 |
from abH and this |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
757 |
have "inv (c \<otimes> a) \<otimes> (c \<otimes> b) \<in> H" by simp |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
758 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
759 |
from carr and this |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
760 |
have "(c \<otimes> b) \<in> (c \<otimes> a) <# H" |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
761 |
by (simp add: lcos_module_rev[OF is_group]) |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
762 |
from carr and this and is_subgroup |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
763 |
show "(c \<otimes> a) <# H = (c \<otimes> b) <# H" by (intro l_repr_independence, simp+) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
764 |
qed |
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19931
diff
changeset
|
765 |
|
14803 | 766 |
|
61382 | 767 |
subsection \<open>Order of a Group and Lagrange's Theorem\<close> |
14803 | 768 |
|
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
769 |
definition |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
770 |
order :: "('a, 'b) monoid_scheme \<Rightarrow> nat" |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
771 |
where "order S = card (carrier S)" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
772 |
|
61628 | 773 |
lemma (in monoid) order_gt_0_iff_finite: "0 < order G \<longleftrightarrow> finite (carrier G)" |
774 |
by(auto simp add: order_def card_gt_0_iff) |
|
775 |
||
14963 | 776 |
lemma (in group) rcosets_part_G: |
27611 | 777 |
assumes "subgroup H G" |
14963 | 778 |
shows "\<Union>(rcosets H) = carrier G" |
27611 | 779 |
proof - |
29237 | 780 |
interpret subgroup H G by fact |
27611 | 781 |
show ?thesis |
68687 | 782 |
unfolding RCOSETS_def r_coset_def by auto |
27611 | 783 |
qed |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
784 |
|
14747 | 785 |
lemma (in group) cosets_finite: |
14963 | 786 |
"\<lbrakk>c \<in> rcosets H; H \<subseteq> carrier G; finite (carrier G)\<rbrakk> \<Longrightarrow> finite c" |
68687 | 787 |
unfolding RCOSETS_def |
788 |
by (auto simp add: r_coset_subset_G [THEN finite_subset]) |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
789 |
|
63167 | 790 |
text\<open>The next two lemmas support the proof of \<open>card_cosets_equal\<close>.\<close> |
14747 | 791 |
lemma (in group) inj_on_f: |
68687 | 792 |
assumes "H \<subseteq> carrier G" and a: "a \<in> carrier G" |
793 |
shows "inj_on (\<lambda>y. y \<otimes> inv a) (H #> a)" |
|
794 |
proof |
|
795 |
fix x y |
|
796 |
assume "x \<in> H #> a" "y \<in> H #> a" and xy: "x \<otimes> inv a = y \<otimes> inv a" |
|
797 |
then have "x \<in> carrier G" "y \<in> carrier G" |
|
798 |
using assms r_coset_subset_G by blast+ |
|
799 |
with xy a show "x = y" |
|
800 |
by auto |
|
801 |
qed |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
802 |
|
14747 | 803 |
lemma (in group) inj_on_g: |
14963 | 804 |
"\<lbrakk>H \<subseteq> carrier G; a \<in> carrier G\<rbrakk> \<Longrightarrow> inj_on (\<lambda>y. y \<otimes> a) H" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
805 |
by (force simp add: inj_on_def subsetD) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
806 |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
807 |
(* ************************************************************************** *) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
808 |
|
14747 | 809 |
lemma (in group) card_cosets_equal: |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
810 |
assumes "R \<in> rcosets H" "H \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
811 |
shows "\<exists>f. bij_betw f H R" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
812 |
proof - |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
813 |
obtain g where g: "g \<in> carrier G" "R = H #> g" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
814 |
using assms(1) unfolding RCOSETS_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
815 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
816 |
let ?f = "\<lambda>h. h \<otimes> g" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
817 |
have "\<And>r. r \<in> R \<Longrightarrow> \<exists>h \<in> H. ?f h = r" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
818 |
proof - |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
819 |
fix r assume "r \<in> R" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
820 |
then obtain h where "h \<in> H" "r = h \<otimes> g" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
821 |
using g unfolding r_coset_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
822 |
thus "\<exists>h \<in> H. ?f h = r" by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
823 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
824 |
hence "R \<subseteq> ?f ` H" by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
825 |
moreover have "?f ` H \<subseteq> R" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
826 |
using g unfolding r_coset_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
827 |
ultimately show ?thesis using inj_on_g unfolding bij_betw_def |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
828 |
using assms(2) g(1) by auto |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
829 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
830 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
831 |
corollary (in group) card_rcosets_equal: |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
832 |
assumes "R \<in> rcosets H" "H \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
833 |
shows "card H = card R" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
834 |
using card_cosets_equal assms bij_betw_same_card by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
835 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
836 |
corollary (in group) rcosets_finite: |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
837 |
assumes "R \<in> rcosets H" "H \<subseteq> carrier G" "finite H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
838 |
shows "finite R" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
839 |
using card_cosets_equal assms bij_betw_finite is_group by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
840 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
841 |
(* ************************************************************************** *) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
842 |
|
14963 | 843 |
lemma (in group) rcosets_subset_PowG: |
844 |
"subgroup H G \<Longrightarrow> rcosets H \<subseteq> Pow(carrier G)" |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
845 |
using rcosets_part_G by auto |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
846 |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
847 |
proposition (in group) lagrange_finite: |
68687 | 848 |
assumes "finite(carrier G)" and HG: "subgroup H G" |
849 |
shows "card(rcosets H) * card(H) = order(G)" |
|
850 |
proof - |
|
851 |
have "card H * card (rcosets H) = card (\<Union>(rcosets H))" |
|
852 |
proof (rule card_partition) |
|
853 |
show "\<And>c1 c2. \<lbrakk>c1 \<in> rcosets H; c2 \<in> rcosets H; c1 \<noteq> c2\<rbrakk> \<Longrightarrow> c1 \<inter> c2 = {}" |
|
68975
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68687
diff
changeset
|
854 |
using HG rcos_disjoint by (auto simp: pairwise_def disjnt_def) |
68687 | 855 |
qed (auto simp: assms finite_UnionD rcosets_part_G card_rcosets_equal subgroup.subset) |
856 |
then show ?thesis |
|
857 |
by (simp add: HG mult.commute order_def rcosets_part_G) |
|
858 |
qed |
|
14803 | 859 |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
860 |
theorem (in group) lagrange: |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
861 |
assumes "subgroup H G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
862 |
shows "card (rcosets H) * card H = order G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
863 |
proof (cases "finite (carrier G)") |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
864 |
case True thus ?thesis using lagrange_finite assms by simp |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
865 |
next |
68687 | 866 |
case False |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
867 |
thus ?thesis |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
868 |
proof (cases "finite H") |
68687 | 869 |
case False thus ?thesis using \<open>infinite (carrier G)\<close> by (simp add: order_def) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
870 |
next |
68687 | 871 |
case True |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
872 |
have "infinite (rcosets H)" |
68687 | 873 |
proof |
874 |
assume "finite (rcosets H)" |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
875 |
hence finite_rcos: "finite (rcosets H)" by simp |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
876 |
hence "card (\<Union>(rcosets H)) = (\<Sum>R\<in>(rcosets H). card R)" |
68687 | 877 |
using card_Union_disjoint[of "rcosets H"] \<open>finite H\<close> rcos_disjoint[OF assms(1)] |
68452
c027dfbfad30
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
paulson <lp15@cam.ac.uk>
parents:
68445
diff
changeset
|
878 |
rcosets_finite[where ?H = H] by (simp add: assms subgroup.subset) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
879 |
hence "order G = (\<Sum>R\<in>(rcosets H). card R)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
880 |
by (simp add: assms order_def rcosets_part_G) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
881 |
hence "order G = (\<Sum>R\<in>(rcosets H). card H)" |
68452
c027dfbfad30
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
paulson <lp15@cam.ac.uk>
parents:
68445
diff
changeset
|
882 |
using card_rcosets_equal by (simp add: assms subgroup.subset) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
883 |
hence "order G = (card H) * (card (rcosets H))" by simp |
68687 | 884 |
hence "order G \<noteq> 0" using finite_rcos \<open>finite H\<close> assms ex_in_conv |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
885 |
rcosets_part_G subgroup.one_closed by fastforce |
68687 | 886 |
thus False using \<open>infinite (carrier G)\<close> order_gt_0_iff_finite by blast |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
887 |
qed |
68687 | 888 |
thus ?thesis using \<open>infinite (carrier G)\<close> by (simp add: order_def) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
889 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
890 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
891 |
|
14803 | 892 |
|
61382 | 893 |
subsection \<open>Quotient Groups: Factorization of a Group\<close> |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
894 |
|
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
895 |
definition |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
896 |
FactGroup :: "[('a,'b) monoid_scheme, 'a set] \<Rightarrow> ('a set) monoid" (infixl "Mod" 65) |
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
67091
diff
changeset
|
897 |
\<comment> \<open>Actually defined for groups rather than monoids\<close> |
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
898 |
where "FactGroup G H = \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, mult = set_mult G, one = H\<rparr>" |
14747 | 899 |
|
14963 | 900 |
lemma (in normal) setmult_closed: |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
901 |
"\<lbrakk>K1 \<in> rcosets H; K2 \<in> rcosets H\<rbrakk> \<Longrightarrow> K1 <#> K2 \<in> rcosets H" |
14963 | 902 |
by (auto simp add: rcos_sum RCOSETS_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
903 |
|
14963 | 904 |
lemma (in normal) setinv_closed: |
905 |
"K \<in> rcosets H \<Longrightarrow> set_inv K \<in> rcosets H" |
|
906 |
by (auto simp add: rcos_inv RCOSETS_def) |
|
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
907 |
|
14963 | 908 |
lemma (in normal) rcosets_assoc: |
909 |
"\<lbrakk>M1 \<in> rcosets H; M2 \<in> rcosets H; M3 \<in> rcosets H\<rbrakk> |
|
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
910 |
\<Longrightarrow> M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
911 |
by (simp add: group.set_mult_assoc is_group rcosets_carrier) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
912 |
|
14963 | 913 |
lemma (in subgroup) subgroup_in_rcosets: |
27611 | 914 |
assumes "group G" |
14963 | 915 |
shows "H \<in> rcosets H" |
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
916 |
proof - |
29237 | 917 |
interpret group G by fact |
26203 | 918 |
from _ subgroup_axioms have "H #> \<one> = H" |
23350 | 919 |
by (rule coset_join2) auto |
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
920 |
then show ?thesis |
14963 | 921 |
by (auto simp add: RCOSETS_def) |
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
922 |
qed |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
923 |
|
14963 | 924 |
lemma (in normal) rcosets_inv_mult_group_eq: |
65035
b46fe5138cb0
backed out unintended effects of 8355a6e2df79 in src/HOL/Algebra
haftmann
parents:
64587
diff
changeset
|
925 |
"M \<in> rcosets H \<Longrightarrow> set_inv M <#> M = H" |
41528 | 926 |
by (auto simp add: RCOSETS_def rcos_inv rcos_sum subgroup.subset normal.axioms normal_axioms) |
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
927 |
|
14963 | 928 |
theorem (in normal) factorgroup_is_group: |
929 |
"group (G Mod H)" |
|
68687 | 930 |
unfolding FactGroup_def |
931 |
apply (rule groupI) |
|
14747 | 932 |
apply (simp add: setmult_closed) |
14963 | 933 |
apply (simp add: normal_imp_subgroup subgroup_in_rcosets [OF is_group]) |
934 |
apply (simp add: restrictI setmult_closed rcosets_assoc) |
|
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
935 |
apply (simp add: normal_imp_subgroup |
14963 | 936 |
subgroup_in_rcosets rcosets_mult_eq) |
937 |
apply (auto dest: rcosets_inv_mult_group_eq simp add: setinv_closed) |
|
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
938 |
done |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
939 |
|
70027
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
940 |
lemma carrier_FactGroup: "carrier(G Mod N) = (\<lambda>x. r_coset G N x) ` carrier G" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
941 |
by (auto simp: FactGroup_def RCOSETS_def) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
942 |
|
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
943 |
lemma one_FactGroup [simp]: "one(G Mod N) = N" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
944 |
by (auto simp: FactGroup_def) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
945 |
|
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
946 |
lemma mult_FactGroup [simp]: "monoid.mult (G Mod N) = set_mult G" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
947 |
by (auto simp: FactGroup_def) |
14803 | 948 |
|
14963 | 949 |
lemma (in normal) inv_FactGroup: |
68687 | 950 |
assumes "X \<in> carrier (G Mod H)" |
951 |
shows "inv\<^bsub>G Mod H\<^esub> X = set_inv X" |
|
952 |
proof - |
|
953 |
have X: "X \<in> rcosets H" |
|
954 |
using assms by (simp add: FactGroup_def) |
|
955 |
moreover have "set_inv X <#> X = H" |
|
956 |
using X by (simp add: normal.rcosets_inv_mult_group_eq normal_axioms) |
|
957 |
moreover have "Group.group (G Mod H)" |
|
958 |
using normal.factorgroup_is_group normal_axioms by blast |
|
959 |
moreover have "set_inv X \<in> rcosets H" |
|
960 |
by (simp add: \<open>X \<in> rcosets H\<close> setinv_closed) |
|
961 |
ultimately show ?thesis |
|
962 |
by (simp add: FactGroup_def group.inv_equality) |
|
963 |
qed |
|
14747 | 964 |
|
69597 | 965 |
text\<open>The coset map is a homomorphism from \<^term>\<open>G\<close> to the quotient group |
966 |
\<^term>\<open>G Mod H\<close>\<close> |
|
14963 | 967 |
lemma (in normal) r_coset_hom_Mod: |
968 |
"(\<lambda>a. H #> a) \<in> hom G (G Mod H)" |
|
969 |
by (auto simp add: FactGroup_def RCOSETS_def Pi_def hom_def rcos_sum) |
|
14747 | 970 |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
971 |
|
70027
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
972 |
lemma (in comm_group) set_mult_commute: |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
973 |
assumes "N \<subseteq> carrier G" "x \<in> rcosets N" "y \<in> rcosets N" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
974 |
shows "x <#> y = y <#> x" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
975 |
using assms unfolding set_mult_def RCOSETS_def |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
976 |
by auto (metis m_comm r_coset_subset_G subsetCE)+ |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
977 |
|
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
978 |
lemma (in comm_group) abelian_FactGroup: |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
979 |
assumes "subgroup N G" shows "comm_group(G Mod N)" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
980 |
proof (rule group.group_comm_groupI) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
981 |
have "N \<lhd> G" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
982 |
by (simp add: assms normal_iff_subgroup) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
983 |
then show "Group.group (G Mod N)" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
984 |
by (simp add: normal.factorgroup_is_group) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
985 |
fix x :: "'a set" and y :: "'a set" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
986 |
assume "x \<in> carrier (G Mod N)" "y \<in> carrier (G Mod N)" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
987 |
then show "x \<otimes>\<^bsub>G Mod N\<^esub> y = y \<otimes>\<^bsub>G Mod N\<^esub> x" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
988 |
apply (simp add: FactGroup_def subgroup_def) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
989 |
apply (rule set_mult_commute) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
990 |
using assms apply (auto simp: subgroup_def) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
991 |
done |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
992 |
qed |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
993 |
|
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
994 |
|
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
995 |
lemma FactGroup_universal: |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
996 |
assumes "h \<in> hom G H" "N \<lhd> G" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
997 |
and h: "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G; r_coset G N x = r_coset G N y\<rbrakk> \<Longrightarrow> h x = h y" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
998 |
obtains g |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
999 |
where "g \<in> hom (G Mod N) H" "\<And>x. x \<in> carrier G \<Longrightarrow> g(r_coset G N x) = h x" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1000 |
proof - |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1001 |
obtain g where g: "\<And>x. x \<in> carrier G \<Longrightarrow> h x = g(r_coset G N x)" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1002 |
using h function_factors_left_gen [of "\<lambda>x. x \<in> carrier G" "r_coset G N" h] by blast |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1003 |
show thesis |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1004 |
proof |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1005 |
show "g \<in> hom (G Mod N) H" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1006 |
proof (rule homI) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1007 |
show "g (u \<otimes>\<^bsub>G Mod N\<^esub> v) = g u \<otimes>\<^bsub>H\<^esub> g v" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1008 |
if "u \<in> carrier (G Mod N)" "v \<in> carrier (G Mod N)" for u v |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1009 |
proof - |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1010 |
from that |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1011 |
obtain x y where xy: "x \<in> carrier G" "u = r_coset G N x" "y \<in> carrier G" "v = r_coset G N y" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1012 |
by (auto simp: carrier_FactGroup) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1013 |
then have "h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1014 |
by (metis hom_mult [OF \<open>h \<in> hom G H\<close>]) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1015 |
then show ?thesis |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1016 |
by (metis Coset.mult_FactGroup xy \<open>N \<lhd> G\<close> g group.subgroup_self normal.axioms(2) normal.rcos_sum subgroup_def) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1017 |
qed |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1018 |
qed (use \<open>h \<in> hom G H\<close> in \<open>auto simp: carrier_FactGroup Pi_iff hom_def simp flip: g\<close>) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1019 |
qed (auto simp flip: g) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1020 |
qed |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1021 |
|
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1022 |
|
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1023 |
lemma (in normal) FactGroup_pow: |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1024 |
fixes k::nat |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1025 |
assumes "a \<in> carrier G" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1026 |
shows "pow (FactGroup G H) (r_coset G H a) k = r_coset G H (pow G a k)" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1027 |
proof (induction k) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1028 |
case 0 |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1029 |
then show ?case |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1030 |
by (simp add: r_coset_def) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1031 |
next |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1032 |
case (Suc k) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1033 |
then show ?case |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1034 |
by (simp add: assms rcos_sum) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1035 |
qed |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1036 |
|
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1037 |
lemma (in normal) FactGroup_int_pow: |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1038 |
fixes k::int |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1039 |
assumes "a \<in> carrier G" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1040 |
shows "pow (FactGroup G H) (r_coset G H a) k = r_coset G H (pow G a k)" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1041 |
by (metis Group.group.axioms(1) image_eqI is_group monoid.nat_pow_closed int_pow_def2 assms |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1042 |
FactGroup_pow carrier_FactGroup inv_FactGroup rcos_inv) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1043 |
|
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1044 |
|
61382 | 1045 |
subsection\<open>The First Isomorphism Theorem\<close> |
14803 | 1046 |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1047 |
text\<open>The quotient by the kernel of a homomorphism is isomorphic to the |
61382 | 1048 |
range of that homomorphism.\<close> |
14803 | 1049 |
|
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
1050 |
definition |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
1051 |
kernel :: "('a, 'm) monoid_scheme \<Rightarrow> ('b, 'n) monoid_scheme \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set" |
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
67091
diff
changeset
|
1052 |
\<comment> \<open>the kernel of a homomorphism\<close> |
67091 | 1053 |
where "kernel G H h = {x. x \<in> carrier G \<and> h x = \<one>\<^bsub>H\<^esub>}" |
14803 | 1054 |
|
1055 |
lemma (in group_hom) subgroup_kernel: "subgroup (kernel G H h) G" |
|
68687 | 1056 |
by (auto simp add: kernel_def group.intro is_group intro: subgroup.intro) |
14803 | 1057 |
|
61382 | 1058 |
text\<open>The kernel of a homomorphism is a normal subgroup\<close> |
14963 | 1059 |
lemma (in group_hom) normal_kernel: "(kernel G H h) \<lhd> G" |
68687 | 1060 |
apply (simp only: G.normal_inv_iff subgroup_kernel) |
1061 |
apply (simp add: kernel_def) |
|
1062 |
done |
|
14803 | 1063 |
|
70039
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1064 |
lemma iso_kernel_image: |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1065 |
assumes "group G" "group H" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1066 |
shows "f \<in> iso G H \<longleftrightarrow> f \<in> hom G H \<and> kernel G H f = {\<one>\<^bsub>G\<^esub>} \<and> f ` carrier G = carrier H" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1067 |
(is "?lhs = ?rhs") |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1068 |
proof (intro iffI conjI) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1069 |
assume f: ?lhs |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1070 |
show "f \<in> hom G H" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1071 |
using Group.iso_iff f by blast |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1072 |
show "kernel G H f = {\<one>\<^bsub>G\<^esub>}" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1073 |
using assms f Group.group_def hom_one |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1074 |
by (fastforce simp add: kernel_def iso_iff_mon_epi mon_iff_hom_one set_eq_iff) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1075 |
show "f ` carrier G = carrier H" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1076 |
by (meson Group.iso_iff f) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1077 |
next |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1078 |
assume ?rhs |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1079 |
with assms show ?lhs |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1080 |
by (auto simp: kernel_def iso_def bij_betw_def inj_on_one_iff') |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1081 |
qed |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1082 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1083 |
|
14803 | 1084 |
lemma (in group_hom) FactGroup_nonempty: |
1085 |
assumes X: "X \<in> carrier (G Mod kernel G H h)" |
|
1086 |
shows "X \<noteq> {}" |
|
1087 |
proof - |
|
1088 |
from X |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1089 |
obtain g where "g \<in> carrier G" |
14803 | 1090 |
and "X = kernel G H h #> g" |
14963 | 1091 |
by (auto simp add: FactGroup_def RCOSETS_def) |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1092 |
thus ?thesis |
14963 | 1093 |
by (auto simp add: kernel_def r_coset_def image_def intro: hom_one) |
14803 | 1094 |
qed |
1095 |
||
1096 |
||
70027
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1097 |
lemma (in group_hom) FactGroup_universal_kernel: |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1098 |
assumes "N \<lhd> G" and h: "N \<subseteq> kernel G H h" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1099 |
obtains g where "g \<in> hom (G Mod N) H" "\<And>x. x \<in> carrier G \<Longrightarrow> g(r_coset G N x) = h x" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1100 |
proof - |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1101 |
have "h x = h y" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1102 |
if "x \<in> carrier G" "y \<in> carrier G" "r_coset G N x = r_coset G N y" for x y |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1103 |
proof - |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1104 |
have "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> y \<in> N" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1105 |
using \<open>N \<lhd> G\<close> group.rcos_self normal.axioms(2) normal_imp_subgroup |
70039
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1106 |
subgroup.rcos_module_imp that by metis |
70027
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1107 |
with h have xy: "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> y \<in> kernel G H h" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1108 |
by blast |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1109 |
have "h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub>(h y) = h (x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> y)" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1110 |
by (simp add: that) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1111 |
also have "\<dots> = \<one>\<^bsub>H\<^esub>" |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1112 |
using xy by (simp add: kernel_def) |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1113 |
finally have "h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub>(h y) = \<one>\<^bsub>H\<^esub>" . |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1114 |
then show ?thesis |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1115 |
using H.inv_equality that by fastforce |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1116 |
qed |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1117 |
with FactGroup_universal [OF homh \<open>N \<lhd> G\<close>] that show thesis |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1118 |
by metis |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1119 |
qed |
94494b92d8d0
some new group theory results: integer group, trivial group, etc.
paulson <lp15@cam.ac.uk>
parents:
70019
diff
changeset
|
1120 |
|
39910 | 1121 |
lemma (in group_hom) FactGroup_the_elem_mem: |
14803 | 1122 |
assumes X: "X \<in> carrier (G Mod (kernel G H h))" |
39910 | 1123 |
shows "the_elem (h`X) \<in> carrier H" |
14803 | 1124 |
proof - |
1125 |
from X |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1126 |
obtain g where g: "g \<in> carrier G" |
14803 | 1127 |
and "X = kernel G H h #> g" |
14963 | 1128 |
by (auto simp add: FactGroup_def RCOSETS_def) |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
61628
diff
changeset
|
1129 |
hence "h ` X = {h g}" by (auto simp add: kernel_def r_coset_def g intro!: imageI) |
14803 | 1130 |
thus ?thesis by (auto simp add: g) |
1131 |
qed |
|
1132 |
||
1133 |
lemma (in group_hom) FactGroup_hom: |
|
39910 | 1134 |
"(\<lambda>X. the_elem (h`X)) \<in> hom (G Mod (kernel G H h)) H" |
68687 | 1135 |
proof - |
1136 |
have "the_elem (h ` (X <#> X')) = the_elem (h ` X) \<otimes>\<^bsub>H\<^esub> the_elem (h ` X')" |
|
1137 |
if X: "X \<in> carrier (G Mod kernel G H h)" and X': "X' \<in> carrier (G Mod kernel G H h)" for X X' |
|
1138 |
proof - |
|
1139 |
obtain g and g' |
|
1140 |
where "g \<in> carrier G" and "g' \<in> carrier G" |
|
1141 |
and "X = kernel G H h #> g" and "X' = kernel G H h #> g'" |
|
1142 |
using X X' by (auto simp add: FactGroup_def RCOSETS_def) |
|
1143 |
hence all: "\<forall>x\<in>X. h x = h g" "\<forall>x\<in>X'. h x = h g'" |
|
1144 |
and Xsub: "X \<subseteq> carrier G" and X'sub: "X' \<subseteq> carrier G" |
|
1145 |
by (force simp add: kernel_def r_coset_def image_def)+ |
|
1146 |
hence "h ` (X <#> X') = {h g \<otimes>\<^bsub>H\<^esub> h g'}" using X X' |
|
1147 |
by (auto dest!: FactGroup_nonempty intro!: image_eqI |
|
1148 |
simp add: set_mult_def |
|
1149 |
subsetD [OF Xsub] subsetD [OF X'sub]) |
|
1150 |
then show "the_elem (h ` (X <#> X')) = the_elem (h ` X) \<otimes>\<^bsub>H\<^esub> the_elem (h ` X')" |
|
1151 |
by (auto simp add: all FactGroup_nonempty X X' the_elem_image_unique) |
|
1152 |
qed |
|
1153 |
then show ?thesis |
|
1154 |
by (simp add: hom_def FactGroup_the_elem_mem normal.factorgroup_is_group [OF normal_kernel] group.axioms monoid.m_closed) |
|
14803 | 1155 |
qed |
1156 |
||
14963 | 1157 |
|
61382 | 1158 |
text\<open>Lemma for the following injectivity result\<close> |
14803 | 1159 |
lemma (in group_hom) FactGroup_subset: |
68687 | 1160 |
assumes "g \<in> carrier G" "g' \<in> carrier G" "h g = h g'" |
1161 |
shows "kernel G H h #> g \<subseteq> kernel G H h #> g'" |
|
1162 |
unfolding kernel_def r_coset_def |
|
1163 |
proof clarsimp |
|
1164 |
fix y |
|
1165 |
assume "y \<in> carrier G" "h y = \<one>\<^bsub>H\<^esub>" |
|
1166 |
with assms show "\<exists>x. x \<in> carrier G \<and> h x = \<one>\<^bsub>H\<^esub> \<and> y \<otimes> g = x \<otimes> g'" |
|
1167 |
by (rule_tac x="y \<otimes> g \<otimes> inv g'" in exI) (auto simp: G.m_assoc) |
|
1168 |
qed |
|
14803 | 1169 |
|
1170 |
lemma (in group_hom) FactGroup_inj_on: |
|
39910 | 1171 |
"inj_on (\<lambda>X. the_elem (h ` X)) (carrier (G Mod kernel G H h))" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1172 |
proof (simp add: inj_on_def, clarify) |
14803 | 1173 |
fix X and X' |
1174 |
assume X: "X \<in> carrier (G Mod kernel G H h)" |
|
1175 |
and X': "X' \<in> carrier (G Mod kernel G H h)" |
|
1176 |
then |
|
1177 |
obtain g and g' |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1178 |
where gX: "g \<in> carrier G" "g' \<in> carrier G" |
14803 | 1179 |
"X = kernel G H h #> g" "X' = kernel G H h #> g'" |
14963 | 1180 |
by (auto simp add: FactGroup_def RCOSETS_def) |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1181 |
hence all: "\<forall>x\<in>X. h x = h g" "\<forall>x\<in>X'. h x = h g'" |
14803 | 1182 |
by (force simp add: kernel_def r_coset_def image_def)+ |
39910 | 1183 |
assume "the_elem (h ` X) = the_elem (h ` X')" |
14803 | 1184 |
hence h: "h g = h g'" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1185 |
by (simp add: all FactGroup_nonempty X X' the_elem_image_unique) |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1186 |
show "X=X'" by (rule equalityI) (simp_all add: FactGroup_subset h gX) |
14803 | 1187 |
qed |
1188 |
||
69597 | 1189 |
text\<open>If the homomorphism \<^term>\<open>h\<close> is onto \<^term>\<open>H\<close>, then so is the |
61382 | 1190 |
homomorphism from the quotient group\<close> |
14803 | 1191 |
lemma (in group_hom) FactGroup_onto: |
1192 |
assumes h: "h ` carrier G = carrier H" |
|
39910 | 1193 |
shows "(\<lambda>X. the_elem (h ` X)) ` carrier (G Mod kernel G H h) = carrier H" |
14803 | 1194 |
proof |
39910 | 1195 |
show "(\<lambda>X. the_elem (h ` X)) ` carrier (G Mod kernel G H h) \<subseteq> carrier H" |
1196 |
by (auto simp add: FactGroup_the_elem_mem) |
|
1197 |
show "carrier H \<subseteq> (\<lambda>X. the_elem (h ` X)) ` carrier (G Mod kernel G H h)" |
|
14803 | 1198 |
proof |
1199 |
fix y |
|
1200 |
assume y: "y \<in> carrier H" |
|
1201 |
with h obtain g where g: "g \<in> carrier G" "h g = y" |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1202 |
by (blast elim: equalityE) |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1203 |
hence "(\<Union>x\<in>kernel G H h #> g. {h x}) = {y}" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1204 |
by (auto simp add: y kernel_def r_coset_def) |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1205 |
with g show "y \<in> (\<lambda>X. the_elem (h ` X)) ` carrier (G Mod kernel G H h)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
61628
diff
changeset
|
1206 |
apply (auto intro!: bexI image_eqI simp add: FactGroup_def RCOSETS_def) |
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
61628
diff
changeset
|
1207 |
apply (subst the_elem_image_unique) |
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
61628
diff
changeset
|
1208 |
apply auto |
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
61628
diff
changeset
|
1209 |
done |
14803 | 1210 |
qed |
1211 |
qed |
|
1212 |
||
1213 |
||
69597 | 1214 |
text\<open>If \<^term>\<open>h\<close> is a homomorphism from \<^term>\<open>G\<close> onto \<^term>\<open>H\<close>, then the |
1215 |
quotient group \<^term>\<open>G Mod (kernel G H h)\<close> is isomorphic to \<^term>\<open>H\<close>.\<close> |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1216 |
theorem (in group_hom) FactGroup_iso_set: |
14803 | 1217 |
"h ` carrier G = carrier H |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1218 |
\<Longrightarrow> (\<lambda>X. the_elem (h`X)) \<in> iso (G Mod (kernel G H h)) H" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1219 |
by (simp add: iso_def FactGroup_hom FactGroup_inj_on bij_betw_def |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1220 |
FactGroup_onto) |
14803 | 1221 |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1222 |
corollary (in group_hom) FactGroup_iso : |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1223 |
"h ` carrier G = carrier H |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1224 |
\<Longrightarrow> (G Mod (kernel G H h))\<cong> H" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1225 |
using FactGroup_iso_set unfolding is_iso_def by auto |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1226 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1227 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
1228 |
lemma (in group_hom) trivial_hom_iff: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close> |
69122
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1229 |
"h ` (carrier G) = { \<one>\<^bsub>H\<^esub> } \<longleftrightarrow> kernel G H h = carrier G" |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1230 |
unfolding kernel_def using one_closed by force |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1231 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
1232 |
lemma (in group_hom) trivial_ker_imp_inj: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1233 |
assumes "kernel G H h = { \<one> }" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1234 |
shows "inj_on h (carrier G)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1235 |
proof (rule inj_onI) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1236 |
fix g1 g2 assume A: "g1 \<in> carrier G" "g2 \<in> carrier G" "h g1 = h g2" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1237 |
hence "h (g1 \<otimes> (inv g2)) = \<one>\<^bsub>H\<^esub>" by simp |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1238 |
hence "g1 \<otimes> (inv g2) = \<one>" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1239 |
using A assms unfolding kernel_def by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1240 |
thus "g1 = g2" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1241 |
using A G.inv_equality G.inv_inv by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1242 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1243 |
|
69122
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1244 |
lemma (in group_hom) inj_iff_trivial_ker: |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1245 |
shows "inj_on h (carrier G) \<longleftrightarrow> kernel G H h = { \<one> }" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1246 |
proof |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1247 |
assume inj: "inj_on h (carrier G)" show "kernel G H h = { \<one> }" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1248 |
unfolding kernel_def |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1249 |
proof (auto) |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1250 |
fix a assume "a \<in> carrier G" "h a = \<one>\<^bsub>H\<^esub>" thus "a = \<one>" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1251 |
using inj hom_one unfolding inj_on_def by force |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1252 |
qed |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1253 |
next |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1254 |
show "kernel G H h = { \<one> } \<Longrightarrow> inj_on h (carrier G)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1255 |
using trivial_ker_imp_inj by simp |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1256 |
qed |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1257 |
|
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1258 |
lemma (in group_hom) induced_group_hom': |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1259 |
assumes "subgroup I G" shows "group_hom (G \<lparr> carrier := I \<rparr>) H h" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1260 |
proof - |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1261 |
have "h \<in> hom (G \<lparr> carrier := I \<rparr>) H" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1262 |
using homh subgroup.subset[OF assms] unfolding hom_def by (auto, meson hom_mult subsetCE) |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1263 |
thus ?thesis |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1264 |
using subgroup.subgroup_is_group[OF assms G.group_axioms] group_axioms |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1265 |
unfolding group_hom_def group_hom_axioms_def by auto |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1266 |
qed |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1267 |
|
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1268 |
lemma (in group_hom) inj_on_subgroup_iff_trivial_ker: |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1269 |
assumes "subgroup I G" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1270 |
shows "inj_on h I \<longleftrightarrow> kernel (G \<lparr> carrier := I \<rparr>) H h = { \<one> }" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1271 |
using group_hom.inj_iff_trivial_ker[OF induced_group_hom'[OF assms]] by simp |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1272 |
|
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1273 |
lemma set_mult_hom: |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1274 |
assumes "h \<in> hom G H" "I \<subseteq> carrier G" and "J \<subseteq> carrier G" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1275 |
shows "h ` (I <#>\<^bsub>G\<^esub> J) = (h ` I) <#>\<^bsub>H\<^esub> (h ` J)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1276 |
proof |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1277 |
show "h ` (I <#>\<^bsub>G\<^esub> J) \<subseteq> (h ` I) <#>\<^bsub>H\<^esub> (h ` J)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1278 |
proof |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1279 |
fix a assume "a \<in> h ` (I <#>\<^bsub>G\<^esub> J)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1280 |
then obtain i j where i: "i \<in> I" and j: "j \<in> J" and "a = h (i \<otimes>\<^bsub>G\<^esub> j)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1281 |
unfolding set_mult_def by auto |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1282 |
hence "a = (h i) \<otimes>\<^bsub>H\<^esub> (h j)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1283 |
using assms unfolding hom_def by blast |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1284 |
thus "a \<in> (h ` I) <#>\<^bsub>H\<^esub> (h ` J)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1285 |
using i and j unfolding set_mult_def by auto |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1286 |
qed |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1287 |
next |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1288 |
show "(h ` I) <#>\<^bsub>H\<^esub> (h ` J) \<subseteq> h ` (I <#>\<^bsub>G\<^esub> J)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1289 |
proof |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1290 |
fix a assume "a \<in> (h ` I) <#>\<^bsub>H\<^esub> (h ` J)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1291 |
then obtain i j where i: "i \<in> I" and j: "j \<in> J" and "a = (h i) \<otimes>\<^bsub>H\<^esub> (h j)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1292 |
unfolding set_mult_def by auto |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1293 |
hence "a = h (i \<otimes>\<^bsub>G\<^esub> j)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1294 |
using assms unfolding hom_def by fastforce |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1295 |
thus "a \<in> h ` (I <#>\<^bsub>G\<^esub> J)" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1296 |
using i and j unfolding set_mult_def by auto |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1297 |
qed |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1298 |
qed |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1299 |
|
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1300 |
corollary coset_hom: |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1301 |
assumes "h \<in> hom G H" "I \<subseteq> carrier G" "a \<in> carrier G" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1302 |
shows "h ` (a <#\<^bsub>G\<^esub> I) = h a <#\<^bsub>H\<^esub> (h ` I)" and "h ` (I #>\<^bsub>G\<^esub> a) = (h ` I) #>\<^bsub>H\<^esub> h a" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1303 |
unfolding l_coset_eq_set_mult r_coset_eq_set_mult using assms set_mult_hom[OF assms(1)] by auto |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1304 |
|
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1305 |
corollary (in group_hom) set_mult_ker_hom: |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1306 |
assumes "I \<subseteq> carrier G" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1307 |
shows "h ` (I <#> (kernel G H h)) = h ` I" and "h ` ((kernel G H h) <#> I) = h ` I" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1308 |
proof - |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1309 |
have ker_in_carrier: "kernel G H h \<subseteq> carrier G" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1310 |
unfolding kernel_def by auto |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1311 |
|
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1312 |
have "h ` (kernel G H h) = { \<one>\<^bsub>H\<^esub> }" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1313 |
unfolding kernel_def by force |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1314 |
moreover have "h ` I \<subseteq> carrier H" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1315 |
using assms by auto |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1316 |
hence "(h ` I) <#>\<^bsub>H\<^esub> { \<one>\<^bsub>H\<^esub> } = h ` I" and "{ \<one>\<^bsub>H\<^esub> } <#>\<^bsub>H\<^esub> (h ` I) = h ` I" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1317 |
unfolding set_mult_def by force+ |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1318 |
ultimately show "h ` (I <#> (kernel G H h)) = h ` I" and "h ` ((kernel G H h) <#> I) = h ` I" |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1319 |
using set_mult_hom[OF homh assms ker_in_carrier] set_mult_hom[OF homh ker_in_carrier assms] by simp+ |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1320 |
qed |
1b5178abaf97
updates to Algebra from Baillon and de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68975
diff
changeset
|
1321 |
|
70039
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1322 |
subsubsection\<open>Trivial homomorphisms\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1323 |
|
70039
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1324 |
definition trivial_homomorphism where |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1325 |
"trivial_homomorphism G H f \<equiv> f \<in> hom G H \<and> (\<forall>x \<in> carrier G. f x = one H)" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1326 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1327 |
lemma trivial_homomorphism_kernel: |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1328 |
"trivial_homomorphism G H f \<longleftrightarrow> f \<in> hom G H \<and> kernel G H f = carrier G" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1329 |
by (auto simp: trivial_homomorphism_def kernel_def) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1330 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1331 |
lemma (in group) trivial_homomorphism_image: |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1332 |
"trivial_homomorphism G H f \<longleftrightarrow> f \<in> hom G H \<and> f ` carrier G = {one H}" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1333 |
by (auto simp: trivial_homomorphism_def) (metis one_closed rev_image_eqI) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1334 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1335 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1336 |
subsection \<open>Image kernel theorems\<close> |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1337 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1338 |
lemma group_Int_image_ker: |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1339 |
assumes f: "f \<in> hom G H" and g: "g \<in> hom H K" and "inj_on (g \<circ> f) (carrier G)" "group G" "group H" "group K" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1340 |
shows "(f ` carrier G) \<inter> (kernel H K g) = {\<one>\<^bsub>H\<^esub>}" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1341 |
proof - |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1342 |
have "(f ` carrier G) \<inter> (kernel H K g) \<subseteq> {\<one>\<^bsub>H\<^esub>}" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1343 |
using assms |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1344 |
apply (clarsimp simp: kernel_def o_def) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1345 |
by (metis group.is_monoid hom_one inj_on_eq_iff monoid.one_closed) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1346 |
moreover have "one H \<in> f ` carrier G" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1347 |
by (metis f \<open>group G\<close> \<open>group H\<close> group.is_monoid hom_one image_iff monoid.one_closed) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1348 |
moreover have "one H \<in> kernel H K g" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1349 |
apply (simp add: kernel_def) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1350 |
using g group.is_monoid hom_one \<open>group H\<close> \<open>group K\<close> by blast |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1351 |
ultimately show ?thesis |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1352 |
by blast |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1353 |
qed |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1354 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1355 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1356 |
lemma group_sum_image_ker: |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1357 |
assumes f: "f \<in> hom G H" and g: "g \<in> hom H K" and eq: "(g \<circ> f) ` (carrier G) = carrier K" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1358 |
and "group G" "group H" "group K" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1359 |
shows "set_mult H (f ` carrier G) (kernel H K g) = carrier H" (is "?lhs = ?rhs") |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1360 |
proof |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1361 |
show "?lhs \<subseteq> ?rhs" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1362 |
apply (auto simp: kernel_def set_mult_def) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1363 |
by (meson Group.group_def assms(5) f hom_carrier image_eqI monoid.m_closed subset_iff) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1364 |
have "\<exists>x\<in>carrier G. \<exists>z. z \<in> carrier H \<and> g z = \<one>\<^bsub>K\<^esub> \<and> y = f x \<otimes>\<^bsub>H\<^esub> z" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1365 |
if y: "y \<in> carrier H" for y |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1366 |
proof - |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1367 |
have "g y \<in> carrier K" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1368 |
using g hom_carrier that by blast |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1369 |
with assms obtain x where x: "x \<in> carrier G" "(g \<circ> f) x = g y" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1370 |
by (metis image_iff) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1371 |
with assms have "inv\<^bsub>H\<^esub> f x \<otimes>\<^bsub>H\<^esub> y \<in> carrier H" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1372 |
by (metis group.subgroup_self hom_carrier image_subset_iff subgroup_def y) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1373 |
moreover |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1374 |
have "g (inv\<^bsub>H\<^esub> f x \<otimes>\<^bsub>H\<^esub> y) = \<one>\<^bsub>K\<^esub>" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1375 |
proof - |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1376 |
have "inv\<^bsub>H\<^esub> f x \<in> carrier H" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1377 |
by (meson \<open>group H\<close> f group.inv_closed hom_carrier image_subset_iff x(1)) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1378 |
then have "g (inv\<^bsub>H\<^esub> f x \<otimes>\<^bsub>H\<^esub> y) = g (inv\<^bsub>H\<^esub> f x) \<otimes>\<^bsub>K\<^esub> g y" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1379 |
by (simp add: hom_mult [OF g] y) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1380 |
also have "\<dots> = inv\<^bsub>K\<^esub> (g (f x)) \<otimes>\<^bsub>K\<^esub> g y" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1381 |
using assms x(1) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1382 |
by (metis (mono_tags, lifting) group_hom.hom_inv group_hom.intro group_hom_axioms.intro hom_carrier image_subset_iff) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1383 |
also have "\<dots> = \<one>\<^bsub>K\<^esub>" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1384 |
using \<open>g y \<in> carrier K\<close> assms(6) group.l_inv x(2) by fastforce |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1385 |
finally show ?thesis . |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1386 |
qed |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1387 |
moreover |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1388 |
have "y = f x \<otimes>\<^bsub>H\<^esub> (inv\<^bsub>H\<^esub> f x \<otimes>\<^bsub>H\<^esub> y)" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1389 |
using x y |
73932
fd21b4a93043
added opaque_combs and renamed hide_lams to opaque_lifting
desharna
parents:
70039
diff
changeset
|
1390 |
by (metis (no_types, opaque_lifting) assms(5) f group.inv_solve_left group.subgroup_self hom_carrier image_subset_iff subgroup_def that) |
70039
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1391 |
ultimately |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1392 |
show ?thesis |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1393 |
using x y by force |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1394 |
qed |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1395 |
then show "?rhs \<subseteq> ?lhs" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1396 |
by (auto simp: kernel_def set_mult_def) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1397 |
qed |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1398 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1399 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1400 |
lemma group_sum_ker_image: |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1401 |
assumes f: "f \<in> hom G H" and g: "g \<in> hom H K" and eq: "(g \<circ> f) ` (carrier G) = carrier K" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1402 |
and "group G" "group H" "group K" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1403 |
shows "set_mult H (kernel H K g) (f ` carrier G) = carrier H" (is "?lhs = ?rhs") |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1404 |
proof |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1405 |
show "?lhs \<subseteq> ?rhs" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1406 |
apply (auto simp: kernel_def set_mult_def) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1407 |
by (meson Group.group_def \<open>group H\<close> f hom_carrier image_eqI monoid.m_closed subset_iff) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1408 |
have "\<exists>w\<in>carrier H. \<exists>x \<in> carrier G. g w = \<one>\<^bsub>K\<^esub> \<and> y = w \<otimes>\<^bsub>H\<^esub> f x" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1409 |
if y: "y \<in> carrier H" for y |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1410 |
proof - |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1411 |
have "g y \<in> carrier K" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1412 |
using g hom_carrier that by blast |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1413 |
with assms obtain x where x: "x \<in> carrier G" "(g \<circ> f) x = g y" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1414 |
by (metis image_iff) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1415 |
with assms have carr: "(y \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> f x) \<in> carrier H" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1416 |
by (metis group.subgroup_self hom_carrier image_subset_iff subgroup_def y) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1417 |
moreover |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1418 |
have "g (y \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> f x) = \<one>\<^bsub>K\<^esub>" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1419 |
proof - |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1420 |
have "inv\<^bsub>H\<^esub> f x \<in> carrier H" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1421 |
by (meson \<open>group H\<close> f group.inv_closed hom_carrier image_subset_iff x(1)) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1422 |
then have "g (y \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> f x) = g y \<otimes>\<^bsub>K\<^esub> g (inv\<^bsub>H\<^esub> f x)" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1423 |
by (simp add: hom_mult [OF g] y) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1424 |
also have "\<dots> = g y \<otimes>\<^bsub>K\<^esub> inv\<^bsub>K\<^esub> (g (f x))" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1425 |
using assms x(1) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1426 |
by (metis (mono_tags, lifting) group_hom.hom_inv group_hom.intro group_hom_axioms.intro hom_carrier image_subset_iff) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1427 |
also have "\<dots> = \<one>\<^bsub>K\<^esub>" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1428 |
using \<open>g y \<in> carrier K\<close> assms(6) group.l_inv x(2) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1429 |
by (simp add: group.r_inv) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1430 |
finally show ?thesis . |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1431 |
qed |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1432 |
moreover |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1433 |
have "y = (y \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> f x) \<otimes>\<^bsub>H\<^esub> f x" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1434 |
using x y by (meson \<open>group H\<close> carr f group.inv_solve_right hom_carrier image_subset_iff) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1435 |
ultimately |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1436 |
show ?thesis |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1437 |
using x y by force |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1438 |
qed |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1439 |
then show "?rhs \<subseteq> ?lhs" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1440 |
by (force simp: kernel_def set_mult_def) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1441 |
qed |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1442 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1443 |
lemma group_semidirect_sum_ker_image: |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1444 |
assumes "(g \<circ> f) \<in> iso G K" "f \<in> hom G H" "g \<in> hom H K" "group G" "group H" "group K" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1445 |
shows "(kernel H K g) \<inter> (f ` carrier G) = {\<one>\<^bsub>H\<^esub>}" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1446 |
"kernel H K g <#>\<^bsub>H\<^esub> (f ` carrier G) = carrier H" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1447 |
using assms |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1448 |
by (simp_all add: iso_iff_mon_epi group_Int_image_ker group_sum_ker_image epi_def mon_def Int_commute [of "kernel H K g"]) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1449 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1450 |
lemma group_semidirect_sum_image_ker: |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1451 |
assumes f: "f \<in> hom G H" and g: "g \<in> hom H K" and iso: "(g \<circ> f) \<in> iso G K" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1452 |
and "group G" "group H" "group K" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1453 |
shows "(f ` carrier G) \<inter> (kernel H K g) = {\<one>\<^bsub>H\<^esub>}" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1454 |
"f ` carrier G <#>\<^bsub>H\<^esub> (kernel H K g) = carrier H" |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1455 |
using group_Int_image_ker [OF f g] group_sum_image_ker [OF f g] assms |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1456 |
by (simp_all add: iso_def bij_betw_def) |
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1457 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1458 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1459 |
|
733e256ecdf3
new group theory material, mostly ported from HOL Light
paulson <lp15@cam.ac.uk>
parents:
70027
diff
changeset
|
1460 |
subsection \<open>Factor Groups and Direct product\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1461 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
1462 |
lemma (in group) DirProd_normal : \<^marker>\<open>contributor \<open>Martin Baillon\<close>\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1463 |
assumes "group K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1464 |
and "H \<lhd> G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1465 |
and "N \<lhd> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1466 |
shows "H \<times> N \<lhd> G \<times>\<times> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1467 |
proof (intro group.normal_invI[OF DirProd_group[OF group_axioms assms(1)]]) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1468 |
show sub : "subgroup (H \<times> N) (G \<times>\<times> K)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1469 |
using DirProd_subgroups[OF group_axioms normal_imp_subgroup[OF assms(2)]assms(1) |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1470 |
normal_imp_subgroup[OF assms(3)]]. |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1471 |
show "\<And>x h. x \<in> carrier (G\<times>\<times>K) \<Longrightarrow> h \<in> H\<times>N \<Longrightarrow> x \<otimes>\<^bsub>G\<times>\<times>K\<^esub> h \<otimes>\<^bsub>G\<times>\<times>K\<^esub> inv\<^bsub>G\<times>\<times>K\<^esub> x \<in> H\<times>N" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1472 |
proof- |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1473 |
fix x h assume xGK : "x \<in> carrier (G \<times>\<times> K)" and hHN : " h \<in> H \<times> N" |
68452
c027dfbfad30
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
paulson <lp15@cam.ac.uk>
parents:
68445
diff
changeset
|
1474 |
hence hGK : "h \<in> carrier (G \<times>\<times> K)" using subgroup.subset[OF sub] by auto |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1475 |
from xGK obtain x1 x2 where x1x2 :"x1 \<in> carrier G" "x2 \<in> carrier K" "x = (x1,x2)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1476 |
unfolding DirProd_def by fastforce |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1477 |
from hHN obtain h1 h2 where h1h2 : "h1 \<in> H" "h2 \<in> N" "h = (h1,h2)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1478 |
unfolding DirProd_def by fastforce |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1479 |
hence h1h2GK : "h1 \<in> carrier G" "h2 \<in> carrier K" |
68687 | 1480 |
using normal_imp_subgroup subgroup.subset assms by blast+ |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1481 |
have "inv\<^bsub>G \<times>\<times> K\<^esub> x = (inv\<^bsub>G\<^esub> x1,inv\<^bsub>K\<^esub> x2)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1482 |
using inv_DirProd[OF group_axioms assms(1) x1x2(1)x1x2(2)] x1x2 by auto |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1483 |
hence "x \<otimes>\<^bsub>G \<times>\<times> K\<^esub> h \<otimes>\<^bsub>G \<times>\<times> K\<^esub> inv\<^bsub>G \<times>\<times> K\<^esub> x = (x1 \<otimes> h1 \<otimes> inv x1,x2 \<otimes>\<^bsub>K\<^esub> h2 \<otimes>\<^bsub>K\<^esub> inv\<^bsub>K\<^esub> x2)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1484 |
using h1h2 x1x2 h1h2GK by auto |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1485 |
moreover have "x1 \<otimes> h1 \<otimes> inv x1 \<in> H" "x2 \<otimes>\<^bsub>K\<^esub> h2 \<otimes>\<^bsub>K\<^esub> inv\<^bsub>K\<^esub> x2 \<in> N" |
68687 | 1486 |
using assms x1x2 h1h2 assms by (simp_all add: normal.inv_op_closed2) |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1487 |
hence "(x1 \<otimes> h1 \<otimes> inv x1, x2 \<otimes>\<^bsub>K\<^esub> h2 \<otimes>\<^bsub>K\<^esub> inv\<^bsub>K\<^esub> x2)\<in> H \<times> N" by auto |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1488 |
ultimately show " x \<otimes>\<^bsub>G \<times>\<times> K\<^esub> h \<otimes>\<^bsub>G \<times>\<times> K\<^esub> inv\<^bsub>G \<times>\<times> K\<^esub> x \<in> H \<times> N" by auto |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1489 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1490 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1491 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
1492 |
lemma (in group) FactGroup_DirProd_multiplication_iso_set : \<^marker>\<open>contributor \<open>Martin Baillon\<close>\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1493 |
assumes "group K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1494 |
and "H \<lhd> G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1495 |
and "N \<lhd> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1496 |
shows "(\<lambda> (X, Y). X \<times> Y) \<in> iso ((G Mod H) \<times>\<times> (K Mod N)) (G \<times>\<times> K Mod H \<times> N)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1497 |
|
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1498 |
proof- |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1499 |
have R :"(\<lambda>(X, Y). X \<times> Y) \<in> carrier (G Mod H) \<times> carrier (K Mod N) \<rightarrow> carrier (G \<times>\<times> K Mod H \<times> N)" |
68687 | 1500 |
unfolding r_coset_def Sigma_def DirProd_def FactGroup_def RCOSETS_def by force |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1501 |
moreover have "(\<forall>x\<in>carrier (G Mod H). \<forall>y\<in>carrier (K Mod N). \<forall>xa\<in>carrier (G Mod H). |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1502 |
\<forall>ya\<in>carrier (K Mod N). (x <#> xa) \<times> (y <#>\<^bsub>K\<^esub> ya) = x \<times> y <#>\<^bsub>G \<times>\<times> K\<^esub> xa \<times> ya)" |
68517 | 1503 |
unfolding set_mult_def by force |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1504 |
moreover have "(\<forall>x\<in>carrier (G Mod H). \<forall>y\<in>carrier (K Mod N). \<forall>xa\<in>carrier (G Mod H). |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1505 |
\<forall>ya\<in>carrier (K Mod N). x \<times> y = xa \<times> ya \<longrightarrow> x = xa \<and> y = ya)" |
68517 | 1506 |
unfolding FactGroup_def using times_eq_iff subgroup.rcosets_non_empty |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1507 |
by (metis assms(2) assms(3) normal_def partial_object.select_convs(1)) |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1508 |
moreover have "(\<lambda>(X, Y). X \<times> Y) ` (carrier (G Mod H) \<times> carrier (K Mod N)) = |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1509 |
carrier (G \<times>\<times> K Mod H \<times> N)" |
68687 | 1510 |
proof - |
1511 |
have 1: "\<And>x a b. \<lbrakk>a \<in> carrier (G Mod H); b \<in> carrier (K Mod N)\<rbrakk> \<Longrightarrow> a \<times> b \<in> carrier (G \<times>\<times> K Mod H \<times> N)" |
|
1512 |
using R by force |
|
1513 |
have 2: "\<And>z. z \<in> carrier (G \<times>\<times> K Mod H \<times> N) \<Longrightarrow> \<exists>x\<in>carrier (G Mod H). \<exists>y\<in>carrier (K Mod N). z = x \<times> y" |
|
1514 |
unfolding DirProd_def FactGroup_def RCOSETS_def r_coset_def by force |
|
1515 |
show ?thesis |
|
1516 |
unfolding image_def by (auto simp: intro: 1 2) |
|
1517 |
qed |
|
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1518 |
ultimately show ?thesis |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1519 |
unfolding iso_def hom_def bij_betw_def inj_on_def by simp |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1520 |
qed |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1521 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
1522 |
corollary (in group) FactGroup_DirProd_multiplication_iso_1 : \<^marker>\<open>contributor \<open>Martin Baillon\<close>\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1523 |
assumes "group K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1524 |
and "H \<lhd> G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1525 |
and "N \<lhd> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1526 |
shows " ((G Mod H) \<times>\<times> (K Mod N)) \<cong> (G \<times>\<times> K Mod H \<times> N)" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1527 |
unfolding is_iso_def using FactGroup_DirProd_multiplication_iso_set assms by auto |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1528 |
|
69895
6b03a8cf092d
more formal contributors (with the help of the history);
wenzelm
parents:
69749
diff
changeset
|
1529 |
corollary (in group) FactGroup_DirProd_multiplication_iso_2 : \<^marker>\<open>contributor \<open>Martin Baillon\<close>\<close> |
68443
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1530 |
assumes "group K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1531 |
and "H \<lhd> G" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1532 |
and "N \<lhd> K" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1533 |
shows "(G \<times>\<times> K Mod H \<times> N) \<cong> ((G Mod H) \<times>\<times> (K Mod N))" |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1534 |
using FactGroup_DirProd_multiplication_iso_1 group.iso_sym assms |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1535 |
DirProd_group[OF normal.factorgroup_is_group normal.factorgroup_is_group] |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1536 |
by blast |
43055b016688
New material from Martin Baillon and Paulo Emílio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1537 |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1538 |
subsubsection "More Lemmas about set multiplication" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1539 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1540 |
(*A group multiplied by a subgroup stays the same*) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1541 |
lemma (in group) set_mult_carrier_idem: |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1542 |
assumes "subgroup H G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1543 |
shows "(carrier G) <#> H = carrier G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1544 |
proof |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1545 |
show "(carrier G)<#>H \<subseteq> carrier G" |
68452
c027dfbfad30
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
paulson <lp15@cam.ac.uk>
parents:
68445
diff
changeset
|
1546 |
unfolding set_mult_def using subgroup.subset assms by blast |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1547 |
next |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1548 |
have " (carrier G) #> \<one> = carrier G" unfolding set_mult_def r_coset_def group_axioms by simp |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1549 |
moreover have "(carrier G) #> \<one> \<subseteq> (carrier G) <#> H" unfolding set_mult_def r_coset_def |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1550 |
using assms subgroup.one_closed[OF assms] by blast |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1551 |
ultimately show "carrier G \<subseteq> (carrier G) <#> H" by simp |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1552 |
qed |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1553 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1554 |
(*Same lemma as above, but everything is included in a subgroup*) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1555 |
lemma (in group) set_mult_subgroup_idem: |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1556 |
assumes HG: "subgroup H G" and NG: "subgroup N (G \<lparr> carrier := H \<rparr>)" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1557 |
shows "H <#> N = H" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1558 |
using group.set_mult_carrier_idem[OF subgroup.subgroup_is_group[OF HG group_axioms] NG] by simp |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1559 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1560 |
(*A normal subgroup is commutative with set_mult*) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1561 |
lemma (in group) commut_normal: |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1562 |
assumes "subgroup H G" and "N\<lhd>G" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1563 |
shows "H<#>N = N<#>H" |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1564 |
proof- |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1565 |
have aux1: "{H <#> N} = {\<Union>h\<in>H. h <# N }" unfolding set_mult_def l_coset_def by auto |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1566 |
also have "... = {\<Union>h\<in>H. N #> h }" using assms normal.coset_eq subgroup.mem_carrier by fastforce |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1567 |
moreover have aux2: "{N <#> H} = {\<Union>h\<in>H. N #> h }"unfolding set_mult_def r_coset_def by auto |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1568 |
ultimately show "H<#>N = N<#>H" by simp |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1569 |
qed |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1570 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1571 |
(*Same lemma as above, but everything is included in a subgroup*) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1572 |
lemma (in group) commut_normal_subgroup: |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1573 |
assumes "subgroup H G" and "N \<lhd> (G\<lparr> carrier := H \<rparr>)" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1574 |
and "subgroup K (G \<lparr> carrier := H \<rparr>)" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1575 |
shows "K <#> N = N <#> K" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1576 |
using group.commut_normal[OF subgroup.subgroup_is_group[OF assms(1) group_axioms] assms(3,2)] by simp |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1577 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1578 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1579 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1580 |
subsubsection "Lemmas about intersection and normal subgroups" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1581 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1582 |
lemma (in group) normal_inter: |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1583 |
assumes "subgroup H G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1584 |
and "subgroup K G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1585 |
and "H1\<lhd>G\<lparr>carrier := H\<rparr>" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1586 |
shows " (H1\<inter>K)\<lhd>(G\<lparr>carrier:= (H\<inter>K)\<rparr>)" |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1587 |
proof- |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1588 |
define HK and H1K and GH and GHK |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1589 |
where "HK = H\<inter>K" and "H1K=H1\<inter>K" and "GH =G\<lparr>carrier := H\<rparr>" and "GHK = (G\<lparr>carrier:= (H\<inter>K)\<rparr>)" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1590 |
show "H1K\<lhd>GHK" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1591 |
proof (intro group.normal_invI[of GHK H1K]) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1592 |
show "Group.group GHK" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1593 |
using GHK_def subgroups_Inter_pair subgroup_imp_group assms by blast |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1594 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1595 |
next |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1596 |
have H1K_incl:"subgroup H1K (G\<lparr>carrier:= (H\<inter>K)\<rparr>)" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1597 |
proof(intro subgroup_incl) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1598 |
show "subgroup H1K G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1599 |
using assms normal_imp_subgroup subgroups_Inter_pair incl_subgroup H1K_def by blast |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1600 |
next |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1601 |
show "subgroup (H\<inter>K) G" using HK_def subgroups_Inter_pair assms by auto |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1602 |
next |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1603 |
have "H1 \<subseteq> (carrier (G\<lparr>carrier:=H\<rparr>))" |
68452
c027dfbfad30
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
paulson <lp15@cam.ac.uk>
parents:
68445
diff
changeset
|
1604 |
using assms(3) normal_imp_subgroup subgroup.subset by blast |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1605 |
also have "... \<subseteq> H" by simp |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1606 |
thus "H1K \<subseteq>H\<inter>K" |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1607 |
using H1K_def calculation by auto |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1608 |
qed |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1609 |
thus "subgroup H1K GHK" using GHK_def by simp |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1610 |
next |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1611 |
show "\<And> x h. x\<in>carrier GHK \<Longrightarrow> h\<in>H1K \<Longrightarrow> x \<otimes>\<^bsub>GHK\<^esub> h \<otimes>\<^bsub>GHK\<^esub> inv\<^bsub>GHK\<^esub> x\<in> H1K" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1612 |
proof- |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1613 |
have invHK: "\<lbrakk>y\<in>HK\<rbrakk> \<Longrightarrow> inv\<^bsub>GHK\<^esub> y = inv\<^bsub>GH\<^esub> y" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1614 |
using m_inv_consistent assms HK_def GH_def GHK_def subgroups_Inter_pair by simp |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1615 |
have multHK : "\<lbrakk>x\<in>HK;y\<in>HK\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>(G\<lparr>carrier:=HK\<rparr>)\<^esub> y = x \<otimes> y" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1616 |
using HK_def by simp |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1617 |
fix x assume p: "x\<in>carrier GHK" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1618 |
fix h assume p2 : "h:H1K" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1619 |
have "carrier(GHK)\<subseteq>HK" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1620 |
using GHK_def HK_def by simp |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1621 |
hence xHK:"x\<in>HK" using p by auto |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1622 |
hence invx:"inv\<^bsub>GHK\<^esub> x = inv\<^bsub>GH\<^esub> x" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1623 |
using invHK assms GHK_def HK_def GH_def m_inv_consistent subgroups_Inter_pair by simp |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1624 |
have "H1\<subseteq>carrier(GH)" |
68452
c027dfbfad30
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
paulson <lp15@cam.ac.uk>
parents:
68445
diff
changeset
|
1625 |
using assms GH_def normal_imp_subgroup subgroup.subset by blast |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1626 |
hence hHK:"h\<in>HK" |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1627 |
using p2 H1K_def HK_def GH_def by auto |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1628 |
hence xhx_egal : "x \<otimes>\<^bsub>GHK\<^esub> h \<otimes>\<^bsub>GHK\<^esub> inv\<^bsub>GHK\<^esub>x = x \<otimes>\<^bsub>GH\<^esub> h \<otimes>\<^bsub>GH\<^esub> inv\<^bsub>GH\<^esub> x" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1629 |
using invx invHK multHK GHK_def GH_def by auto |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1630 |
have xH:"x\<in>carrier(GH)" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1631 |
using xHK HK_def GH_def by auto |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1632 |
have hH:"h\<in>carrier(GH)" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1633 |
using hHK HK_def GH_def by auto |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1634 |
have "(\<forall>x\<in>carrier (GH). \<forall>h\<in>H1. x \<otimes>\<^bsub>GH\<^esub> h \<otimes>\<^bsub>GH\<^esub> inv\<^bsub>GH\<^esub> x \<in> H1)" |
68687 | 1635 |
using assms GH_def normal.inv_op_closed2 by fastforce |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1636 |
hence INCL_1 : "x \<otimes>\<^bsub>GH\<^esub> h \<otimes>\<^bsub>GH\<^esub> inv\<^bsub>GH\<^esub> x \<in> H1" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1637 |
using xH H1K_def p2 by blast |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1638 |
have " x \<otimes>\<^bsub>GH\<^esub> h \<otimes>\<^bsub>GH\<^esub> inv\<^bsub>GH\<^esub> x \<in> HK" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1639 |
using assms HK_def subgroups_Inter_pair hHK xHK |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1640 |
by (metis GH_def inf.cobounded1 subgroup_def subgroup_incl) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1641 |
hence " x \<otimes>\<^bsub>GH\<^esub> h \<otimes>\<^bsub>GH\<^esub> inv\<^bsub>GH\<^esub> x \<in> K" using HK_def by simp |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1642 |
hence " x \<otimes>\<^bsub>GH\<^esub> h \<otimes>\<^bsub>GH\<^esub> inv\<^bsub>GH\<^esub> x \<in> H1K" using INCL_1 H1K_def by auto |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1643 |
thus "x \<otimes>\<^bsub>GHK\<^esub> h \<otimes>\<^bsub>GHK\<^esub> inv\<^bsub>GHK\<^esub> x \<in> H1K" using xhx_egal by simp |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1644 |
qed |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1645 |
qed |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1646 |
qed |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1647 |
|
70019
095dce9892e8
A few results in Algebra, and bits for Analysis
paulson <lp15@cam.ac.uk>
parents:
69895
diff
changeset
|
1648 |
lemma (in group) normal_Int_subgroup: |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1649 |
assumes "subgroup H G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1650 |
and "N \<lhd> G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1651 |
shows "(N\<inter>H) \<lhd> (G\<lparr>carrier := H\<rparr>)" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1652 |
proof - |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1653 |
define K where "K = carrier G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1654 |
have "G\<lparr>carrier := K\<rparr> = G" using K_def by auto |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1655 |
moreover have "subgroup K G" using K_def subgroup_self by blast |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1656 |
moreover have "normal N (G \<lparr>carrier :=K\<rparr>)" using assms K_def by simp |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1657 |
ultimately have "N \<inter> H \<lhd> G\<lparr>carrier := K \<inter> H\<rparr>" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1658 |
using normal_inter[of K H N] assms(1) by blast |
68452
c027dfbfad30
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
paulson <lp15@cam.ac.uk>
parents:
68445
diff
changeset
|
1659 |
moreover have "K \<inter> H = H" using K_def assms subgroup.subset by blast |
68687 | 1660 |
ultimately show "normal (N\<inter>H) (G\<lparr>carrier := H\<rparr>)" |
1661 |
by auto |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1662 |
qed |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1663 |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
1664 |
end |