src/HOL/Algebra/RingHom.thy
 author wenzelm Thu Jun 21 20:07:26 2007 +0200 (2007-06-21) changeset 23464 bc2563c37b1a parent 23463 9953ff53cc64 child 26204 da9778392d8c permissions -rw-r--r--
tuned proofs -- avoid implicit prems;
```     1 (*
```
```     2   Title:     HOL/Algebra/RingHom.thy
```
```     3   Id:        \$Id\$
```
```     4   Author:    Stephan Hohe, TU Muenchen
```
```     5 *)
```
```     6
```
```     7 theory RingHom
```
```     8 imports Ideal
```
```     9 begin
```
```    10
```
```    11 section {* Homomorphisms of Non-Commutative Rings *}
```
```    12
```
```    13 text {* Lifting existing lemmas in a @{text ring_hom_ring} locale *}
```
```    14 locale ring_hom_ring = ring R + ring S + var h +
```
```    15   assumes homh: "h \<in> ring_hom R S"
```
```    16   notes hom_mult [simp] = ring_hom_mult [OF homh]
```
```    17     and hom_one [simp] = ring_hom_one [OF homh]
```
```    18
```
```    19 interpretation ring_hom_cring \<subseteq> ring_hom_ring
```
```    20   by (unfold_locales, rule homh)
```
```    21
```
```    22 interpretation ring_hom_ring \<subseteq> abelian_group_hom R S
```
```    23 apply (rule abelian_group_homI)
```
```    24   apply (rule R.is_abelian_group)
```
```    25  apply (rule S.is_abelian_group)
```
```    26 apply (intro group_hom.intro group_hom_axioms.intro)
```
```    27   apply (rule R.a_group)
```
```    28  apply (rule S.a_group)
```
```    29 apply (insert homh, unfold hom_def ring_hom_def)
```
```    30 apply simp
```
```    31 done
```
```    32
```
```    33 lemma (in ring_hom_ring) is_ring_hom_ring:
```
```    34   includes struct R + struct S
```
```    35   shows "ring_hom_ring R S h"
```
```    36 by fact
```
```    37
```
```    38 lemma ring_hom_ringI:
```
```    39   includes ring R + ring S
```
```    40   assumes (* morphism: "h \<in> carrier R \<rightarrow> carrier S" *)
```
```    41           hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
```
```    42       and compatible_mult: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
```
```    43       and compatible_add: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
```
```    44       and compatible_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
```
```    45   shows "ring_hom_ring R S h"
```
```    46 apply unfold_locales
```
```    47 apply (unfold ring_hom_def, safe)
```
```    48    apply (simp add: hom_closed Pi_def)
```
```    49   apply (erule (1) compatible_mult)
```
```    50  apply (erule (1) compatible_add)
```
```    51 apply (rule compatible_one)
```
```    52 done
```
```    53
```
```    54 lemma ring_hom_ringI2:
```
```    55   includes ring R + ring S
```
```    56   assumes h: "h \<in> ring_hom R S"
```
```    57   shows "ring_hom_ring R S h"
```
```    58 apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro)
```
```    59 apply (rule R.is_ring)
```
```    60 apply (rule S.is_ring)
```
```    61 apply (rule h)
```
```    62 done
```
```    63
```
```    64 lemma ring_hom_ringI3:
```
```    65   includes abelian_group_hom R S + ring R + ring S
```
```    66   assumes compatible_mult: "!!x y. [| x : carrier R; y : carrier R |] ==> h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
```
```    67       and compatible_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
```
```    68   shows "ring_hom_ring R S h"
```
```    69 apply (intro ring_hom_ring.intro ring_hom_ring_axioms.intro, rule R.is_ring, rule S.is_ring)
```
```    70 apply (insert group_hom.homh[OF a_group_hom])
```
```    71 apply (unfold hom_def ring_hom_def, simp)
```
```    72 apply safe
```
```    73 apply (erule (1) compatible_mult)
```
```    74 apply (rule compatible_one)
```
```    75 done
```
```    76
```
```    77 lemma ring_hom_cringI:
```
```    78   includes ring_hom_ring R S h + cring R + cring S
```
```    79   shows "ring_hom_cring R S h"
```
```    80   by (intro ring_hom_cring.intro ring_hom_cring_axioms.intro)
```
```    81     (rule R.is_cring, rule S.is_cring, rule homh)
```
```    82
```
```    83
```
```    84 subsection {* The kernel of a ring homomorphism *}
```
```    85
```
```    86 --"the kernel of a ring homomorphism is an ideal"
```
```    87 lemma (in ring_hom_ring) kernel_is_ideal:
```
```    88   shows "ideal (a_kernel R S h) R"
```
```    89 apply (rule idealI)
```
```    90    apply (rule R.is_ring)
```
```    91   apply (rule additive_subgroup.a_subgroup[OF additive_subgroup_a_kernel])
```
```    92  apply (unfold a_kernel_def', simp+)
```
```    93 done
```
```    94
```
```    95 text {* Elements of the kernel are mapped to zero *}
```
```    96 lemma (in abelian_group_hom) kernel_zero [simp]:
```
```    97   "i \<in> a_kernel R S h \<Longrightarrow> h i = \<zero>\<^bsub>S\<^esub>"
```
```    98 by (simp add: a_kernel_defs)
```
```    99
```
```   100
```
```   101 subsection {* Cosets *}
```
```   102
```
```   103 text {* Cosets of the kernel correspond to the elements of the image of the homomorphism *}
```
```   104 lemma (in ring_hom_ring) rcos_imp_homeq:
```
```   105   assumes acarr: "a \<in> carrier R"
```
```   106       and xrcos: "x \<in> a_kernel R S h +> a"
```
```   107   shows "h x = h a"
```
```   108 proof -
```
```   109   interpret ideal ["a_kernel R S h" "R"] by (rule kernel_is_ideal)
```
```   110
```
```   111   from xrcos
```
```   112       have "\<exists>i \<in> a_kernel R S h. x = i \<oplus> a" by (simp add: a_r_coset_defs)
```
```   113   from this obtain i
```
```   114       where iker: "i \<in> a_kernel R S h"
```
```   115         and x: "x = i \<oplus> a"
```
```   116       by fast+
```
```   117   note carr = acarr iker[THEN a_Hcarr]
```
```   118
```
```   119   from x
```
```   120       have "h x = h (i \<oplus> a)" by simp
```
```   121   also from carr
```
```   122       have "\<dots> = h i \<oplus>\<^bsub>S\<^esub> h a" by simp
```
```   123   also from iker
```
```   124       have "\<dots> = \<zero>\<^bsub>S\<^esub> \<oplus>\<^bsub>S\<^esub> h a" by simp
```
```   125   also from carr
```
```   126       have "\<dots> = h a" by simp
```
```   127   finally
```
```   128       show "h x = h a" .
```
```   129 qed
```
```   130
```
```   131 lemma (in ring_hom_ring) homeq_imp_rcos:
```
```   132   assumes acarr: "a \<in> carrier R"
```
```   133       and xcarr: "x \<in> carrier R"
```
```   134       and hx: "h x = h a"
```
```   135   shows "x \<in> a_kernel R S h +> a"
```
```   136 proof -
```
```   137   interpret ideal ["a_kernel R S h" "R"] by (rule kernel_is_ideal)
```
```   138
```
```   139   note carr = acarr xcarr
```
```   140   note hcarr = acarr[THEN hom_closed] xcarr[THEN hom_closed]
```
```   141
```
```   142   from hx and hcarr
```
```   143       have a: "h x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub>h a = \<zero>\<^bsub>S\<^esub>" by algebra
```
```   144   from carr
```
```   145       have "h x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub>h a = h (x \<oplus> \<ominus>a)" by simp
```
```   146   from a and this
```
```   147       have b: "h (x \<oplus> \<ominus>a) = \<zero>\<^bsub>S\<^esub>" by simp
```
```   148
```
```   149   from carr have "x \<oplus> \<ominus>a \<in> carrier R" by simp
```
```   150   from this and b
```
```   151       have "x \<oplus> \<ominus>a \<in> a_kernel R S h"
```
```   152       unfolding a_kernel_def'
```
```   153       by fast
```
```   154
```
```   155   from this and carr
```
```   156       show "x \<in> a_kernel R S h +> a" by (simp add: a_rcos_module_rev)
```
```   157 qed
```
```   158
```
```   159 corollary (in ring_hom_ring) rcos_eq_homeq:
```
```   160   assumes acarr: "a \<in> carrier R"
```
```   161   shows "(a_kernel R S h) +> a = {x \<in> carrier R. h x = h a}"
```
```   162 apply rule defer 1
```
```   163 apply clarsimp defer 1
```
```   164 proof
```
```   165   interpret ideal ["a_kernel R S h" "R"] by (rule kernel_is_ideal)
```
```   166
```
```   167   fix x
```
```   168   assume xrcos: "x \<in> a_kernel R S h +> a"
```
```   169   from acarr and this
```
```   170       have xcarr: "x \<in> carrier R"
```
```   171       by (rule a_elemrcos_carrier)
```
```   172
```
```   173   from xrcos
```
```   174       have "h x = h a" by (rule rcos_imp_homeq[OF acarr])
```
```   175   from xcarr and this
```
```   176       show "x \<in> {x \<in> carrier R. h x = h a}" by fast
```
```   177 next
```
```   178   interpret ideal ["a_kernel R S h" "R"] by (rule kernel_is_ideal)
```
```   179
```
```   180   fix x
```
```   181   assume xcarr: "x \<in> carrier R"
```
```   182      and hx: "h x = h a"
```
```   183   from acarr xcarr hx
```
```   184       show "x \<in> a_kernel R S h +> a" by (rule homeq_imp_rcos)
```
```   185 qed
```
```   186
```
```   187 end
```