author  wenzelm 
Thu, 14 Jun 2007 10:38:48 +0200  
changeset 23383  5460951833fa 
parent 23350  50c5b0912a0c 
child 23463  9953ff53cc64 
permissions  rwrr 
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(* 
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Title: HOL/Algebra/AbelCoset.thy 
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Id: $Id$ 
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Author: Stephan Hohe, TU Muenchen 
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*) 
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theory AbelCoset 
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imports Coset Ring 
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begin 
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section {* More Lifting from Groups to Abelian Groups *} 
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subsection {* Definitions *} 
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21502  16 
text {* Hiding @{text "<+>"} from @{theory Sum_Type} until I come 
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up with better syntax here *} 
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hide const Plus 
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constdefs (structure G) 
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a_r_coset :: "[_, 'a set, 'a] \<Rightarrow> 'a set" (infixl "+>\<index>" 60) 
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"a_r_coset G \<equiv> r_coset \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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a_l_coset :: "[_, 'a, 'a set] \<Rightarrow> 'a set" (infixl "<+\<index>" 60) 
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"a_l_coset G \<equiv> l_coset \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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A_RCOSETS :: "[_, 'a set] \<Rightarrow> ('a set)set" ("a'_rcosets\<index> _" [81] 80) 
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"A_RCOSETS G H \<equiv> RCOSETS \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H" 
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set_add :: "[_, 'a set ,'a set] \<Rightarrow> 'a set" (infixl "<+>\<index>" 60) 
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"set_add G \<equiv> set_mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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A_SET_INV :: "[_,'a set] \<Rightarrow> 'a set" ("a'_set'_inv\<index> _" [81] 80) 
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"A_SET_INV G H \<equiv> SET_INV \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H" 
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constdefs (structure G) 
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a_r_congruent :: "[('a,'b)ring_scheme, 'a set] \<Rightarrow> ('a*'a)set" 
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("racong\<index> _") 
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"a_r_congruent G \<equiv> r_congruent \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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constdefs 
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A_FactGroup :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) monoid" 
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(infixl "A'_Mod" 65) 
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{*Actually defined for groups rather than monoids*} 
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"A_FactGroup G H \<equiv> FactGroup \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H" 
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constdefs 
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a_kernel :: "('a, 'm) ring_scheme \<Rightarrow> ('b, 'n) ring_scheme \<Rightarrow> 
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('a \<Rightarrow> 'b) \<Rightarrow> 'a set" 
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{*the kernel of a homomorphism (additive)*} 
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"a_kernel G H h \<equiv> kernel \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> 
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\<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr> h" 
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locale abelian_group_hom = abelian_group G + abelian_group H + var h + 
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assumes a_group_hom: "group_hom ( carrier = carrier G, mult = add G, one = zero G ) 
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( carrier = carrier H, mult = add H, one = zero H ) h" 
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lemmas a_r_coset_defs = 
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a_r_coset_def r_coset_def 
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lemma a_r_coset_def': 
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includes struct G 
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shows "H +> a \<equiv> \<Union>h\<in>H. {h \<oplus> a}" 
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unfolding a_r_coset_defs 
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by simp 
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lemmas a_l_coset_defs = 
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a_l_coset_def l_coset_def 
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lemma a_l_coset_def': 
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includes struct G 
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shows "a <+ H \<equiv> \<Union>h\<in>H. {a \<oplus> h}" 
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unfolding a_l_coset_defs 
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by simp 
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lemmas A_RCOSETS_defs = 
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A_RCOSETS_def RCOSETS_def 
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lemma A_RCOSETS_def': 
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includes struct G 
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shows "a_rcosets H \<equiv> \<Union>a\<in>carrier G. {H +> a}" 
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unfolding A_RCOSETS_defs 
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by (fold a_r_coset_def, simp) 
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lemmas set_add_defs = 
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set_add_def set_mult_def 
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lemma set_add_def': 
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includes struct G 
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shows "H <+> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<oplus> k}" 
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unfolding set_add_defs 
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by simp 
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lemmas A_SET_INV_defs = 
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A_SET_INV_def SET_INV_def 
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lemma A_SET_INV_def': 
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includes struct G 
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shows "a_set_inv H \<equiv> \<Union>h\<in>H. {\<ominus> h}" 
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unfolding A_SET_INV_defs 
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by (fold a_inv_def) 
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subsection {* Cosets *} 
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lemma (in abelian_group) a_coset_add_assoc: 
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"[ M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G ] 
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==> (M +> g) +> h = M +> (g \<oplus> h)" 
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by (rule group.coset_mult_assoc [OF a_group, 
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folded a_r_coset_def, simplified monoid_record_simps]) 
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lemma (in abelian_group) a_coset_add_zero [simp]: 
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"M \<subseteq> carrier G ==> M +> \<zero> = M" 
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by (rule group.coset_mult_one [OF a_group, 
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folded a_r_coset_def, simplified monoid_record_simps]) 
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lemma (in abelian_group) a_coset_add_inv1: 
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"[ M +> (x \<oplus> (\<ominus> y)) = M; x \<in> carrier G ; y \<in> carrier G; 
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M \<subseteq> carrier G ] ==> M +> x = M +> y" 
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by (rule group.coset_mult_inv1 [OF a_group, 
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folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) 
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lemma (in abelian_group) a_coset_add_inv2: 
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"[ M +> x = M +> y; x \<in> carrier G; y \<in> carrier G; M \<subseteq> carrier G ] 
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==> M +> (x \<oplus> (\<ominus> y)) = M" 
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by (rule group.coset_mult_inv2 [OF a_group, 
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folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) 
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lemma (in abelian_group) a_coset_join1: 
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"[ H +> x = H; x \<in> carrier G; subgroup H (carrier = carrier G, mult = add G, one = zero G) ] ==> x \<in> H" 
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by (rule group.coset_join1 [OF a_group, 
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folded a_r_coset_def, simplified monoid_record_simps]) 
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lemma (in abelian_group) a_solve_equation: 
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"\<lbrakk>subgroup H (carrier = carrier G, mult = add G, one = zero G); x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<oplus> x" 
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by (rule group.solve_equation [OF a_group, 
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folded a_r_coset_def, simplified monoid_record_simps]) 
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lemma (in abelian_group) a_repr_independence: 
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"\<lbrakk>y \<in> H +> x; x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> \<rbrakk> \<Longrightarrow> H +> x = H +> y" 
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by (rule group.repr_independence [OF a_group, 
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folded a_r_coset_def, simplified monoid_record_simps]) 
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145 
lemma (in abelian_group) a_coset_join2: 
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"\<lbrakk>x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>; x\<in>H\<rbrakk> \<Longrightarrow> H +> x = H" 
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by (rule group.coset_join2 [OF a_group, 
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folded a_r_coset_def, simplified monoid_record_simps]) 
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149 

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lemma (in abelian_monoid) a_r_coset_subset_G: 
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"[ H \<subseteq> carrier G; x \<in> carrier G ] ==> H +> x \<subseteq> carrier G" 
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by (rule monoid.r_coset_subset_G [OF a_monoid, 
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folded a_r_coset_def, simplified monoid_record_simps]) 
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154 

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lemma (in abelian_group) a_rcosI: 
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"[ h \<in> H; H \<subseteq> carrier G; x \<in> carrier G] ==> h \<oplus> x \<in> H +> x" 
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by (rule group.rcosI [OF a_group, 
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folded a_r_coset_def, simplified monoid_record_simps]) 
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159 

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lemma (in abelian_group) a_rcosetsI: 
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"\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H +> x \<in> a_rcosets H" 
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by (rule group.rcosetsI [OF a_group, 
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folded a_r_coset_def A_RCOSETS_def, simplified monoid_record_simps]) 
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164 

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text{*Really needed?*} 
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lemma (in abelian_group) a_transpose_inv: 
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"[ x \<oplus> y = z; x \<in> carrier G; y \<in> carrier G; z \<in> carrier G ] 
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==> (\<ominus> x) \<oplus> z = y" 
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by (rule group.transpose_inv [OF a_group, 
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folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) 
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(* 
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"duplicate" 
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lemma (in abelian_group) a_rcos_self: 
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"[ x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> ] ==> x \<in> H +> x" 
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by (rule group.rcos_self [OF a_group, 
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folded a_r_coset_def, simplified monoid_record_simps]) 
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*) 
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180 

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subsection {* Subgroups *} 
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locale additive_subgroup = var H + struct G + 
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assumes a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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185 

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lemma (in additive_subgroup) is_additive_subgroup: 
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shows "additive_subgroup H G" 
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by fact 
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189 

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lemma additive_subgroupI: 
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includes struct G 
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assumes a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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shows "additive_subgroup H G" 
23350  194 
by (rule additive_subgroup.intro) (rule a_subgroup) 
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195 

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lemma (in additive_subgroup) a_subset: 
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"H \<subseteq> carrier G" 
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by (rule subgroup.subset[OF a_subgroup, 
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simplified monoid_record_simps]) 
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200 

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lemma (in additive_subgroup) a_closed [intro, simp]: 
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"\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> y \<in> H" 
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by (rule subgroup.m_closed[OF a_subgroup, 
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simplified monoid_record_simps]) 
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205 

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lemma (in additive_subgroup) zero_closed [simp]: 
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"\<zero> \<in> H" 
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by (rule subgroup.one_closed[OF a_subgroup, 
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simplified monoid_record_simps]) 
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210 

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lemma (in additive_subgroup) a_inv_closed [intro,simp]: 
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"x \<in> H \<Longrightarrow> \<ominus> x \<in> H" 
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by (rule subgroup.m_inv_closed[OF a_subgroup, 
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folded a_inv_def, simplified monoid_record_simps]) 
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215 

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216 

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217 
subsection {* Normal additive subgroups *} 
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218 

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219 
subsubsection {* Definition of @{text "abelian_subgroup"} *} 
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220 

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text {* Every subgroup of an @{text "abelian_group"} is normal *} 
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222 

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locale abelian_subgroup = additive_subgroup H G + abelian_group G + 
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assumes a_normal: "normal H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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225 

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226 
lemma (in abelian_subgroup) is_abelian_subgroup: 
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227 
shows "abelian_subgroup H G" 
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by fact 
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229 

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230 
lemma abelian_subgroupI: 
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231 
assumes a_normal: "normal H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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232 
and a_comm: "!!x y. [ x \<in> carrier G; y \<in> carrier G ] ==> x \<oplus>\<^bsub>G\<^esub> y = y \<oplus>\<^bsub>G\<^esub> x" 
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233 
shows "abelian_subgroup H G" 
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234 
proof  
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235 
interpret normal ["H" "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"] 
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236 
by (rule a_normal) 
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237 

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238 
show "abelian_subgroup H G" 
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239 
by (unfold_locales, simp add: a_comm) 
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240 
qed 
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241 

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242 
lemma abelian_subgroupI2: 
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243 
includes struct G 
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244 
assumes a_comm_group: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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245 
and a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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246 
shows "abelian_subgroup H G" 
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247 
proof  
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248 
interpret comm_group ["\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"] 
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249 
by (rule a_comm_group) 
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250 
interpret subgroup ["H" "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"] 
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251 
by (rule a_subgroup) 
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252 

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253 
show "abelian_subgroup H G" 
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254 
apply unfold_locales 
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255 
proof (simp add: r_coset_def l_coset_def, clarsimp) 
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256 
fix x 
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assume xcarr: "x \<in> carrier G" 
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258 
from a_subgroup 
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have Hcarr: "H \<subseteq> carrier G" by (unfold subgroup_def, simp) 
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260 
from xcarr Hcarr 
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261 
show "(\<Union>h\<in>H. {h \<oplus>\<^bsub>G\<^esub> x}) = (\<Union>h\<in>H. {x \<oplus>\<^bsub>G\<^esub> h})" 
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262 
using m_comm[simplified] 
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263 
by fast 
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264 
qed 
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265 
qed 
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266 

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267 
lemma abelian_subgroupI3: 
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268 
includes struct G 
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assumes asg: "additive_subgroup H G" 
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270 
and ag: "abelian_group G" 
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271 
shows "abelian_subgroup H G" 
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apply (rule abelian_subgroupI2) 
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apply (rule abelian_group.a_comm_group[OF ag]) 
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apply (rule additive_subgroup.a_subgroup[OF asg]) 
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275 
done 
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276 

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277 
lemma (in abelian_subgroup) a_coset_eq: 
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"(\<forall>x \<in> carrier G. H +> x = x <+ H)" 
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279 
by (rule normal.coset_eq[OF a_normal, 
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folded a_r_coset_def a_l_coset_def, simplified monoid_record_simps]) 
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281 

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282 
lemma (in abelian_subgroup) a_inv_op_closed1: 
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283 
shows "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (\<ominus> x) \<oplus> h \<oplus> x \<in> H" 
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284 
by (rule normal.inv_op_closed1 [OF a_normal, 
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285 
folded a_inv_def, simplified monoid_record_simps]) 
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286 

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lemma (in abelian_subgroup) a_inv_op_closed2: 
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288 
shows "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> h \<oplus> (\<ominus> x) \<in> H" 
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289 
by (rule normal.inv_op_closed2 [OF a_normal, 
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290 
folded a_inv_def, simplified monoid_record_simps]) 
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291 

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292 
text{*Alternative characterization of normal subgroups*} 
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293 
lemma (in abelian_group) a_normal_inv_iff: 
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Restructured algebra library, added ideals and quotient rings.
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parents:
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294 
"(N \<lhd> \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>) = 
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Restructured algebra library, added ideals and quotient rings.
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parents:
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295 
(subgroup N \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> & (\<forall>x \<in> carrier G. \<forall>h \<in> N. x \<oplus> h \<oplus> (\<ominus> x) \<in> N))" 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

296 
(is "_ = ?rhs") 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

297 
by (rule group.normal_inv_iff [OF a_group, 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

298 
folded a_inv_def, simplified monoid_record_simps]) 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

299 

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Restructured algebra library, added ideals and quotient rings.
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parents:
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300 
lemma (in abelian_group) a_lcos_m_assoc: 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

301 
"[ M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G ] 
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Restructured algebra library, added ideals and quotient rings.
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parents:
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302 
==> g <+ (h <+ M) = (g \<oplus> h) <+ M" 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

303 
by (rule group.lcos_m_assoc [OF a_group, 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

304 
folded a_l_coset_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

305 

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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
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changeset

306 
lemma (in abelian_group) a_lcos_mult_one: 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

307 
"M \<subseteq> carrier G ==> \<zero> <+ M = M" 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

308 
by (rule group.lcos_mult_one [OF a_group, 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

309 
folded a_l_coset_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

310 

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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

311 

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Restructured algebra library, added ideals and quotient rings.
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changeset

312 
lemma (in abelian_group) a_l_coset_subset_G: 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

313 
"[ H \<subseteq> carrier G; x \<in> carrier G ] ==> x <+ H \<subseteq> carrier G" 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

314 
by (rule group.l_coset_subset_G [OF a_group, 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

315 
folded a_l_coset_def, simplified monoid_record_simps]) 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

316 

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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

317 

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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

318 
lemma (in abelian_group) a_l_coset_swap: 
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

319 
"\<lbrakk>y \<in> x <+ H; x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>\<rbrakk> \<Longrightarrow> x \<in> y <+ H" 
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

320 
by (rule group.l_coset_swap [OF a_group, 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

321 
folded a_l_coset_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

322 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

323 
lemma (in abelian_group) a_l_coset_carrier: 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

324 
"[ y \<in> x <+ H; x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> ] ==> y \<in> carrier G" 
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

325 
by (rule group.l_coset_carrier [OF a_group, 
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

326 
folded a_l_coset_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

327 

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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

328 
lemma (in abelian_group) a_l_repr_imp_subset: 
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

329 
assumes y: "y \<in> x <+ H" and x: "x \<in> carrier G" and sb: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

330 
shows "y <+ H \<subseteq> x <+ H" 
23350  331 
apply (rule group.l_repr_imp_subset [OF a_group, 
20318
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

332 
folded a_l_coset_def, simplified monoid_record_simps]) 
23350  333 
apply (rule y) 
334 
apply (rule x) 

335 
apply (rule sb) 

336 
done 

20318
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

337 

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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

338 
lemma (in abelian_group) a_l_repr_independence: 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

339 
assumes y: "y \<in> x <+ H" and x: "x \<in> carrier G" and sb: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

340 
shows "x <+ H = y <+ H" 
23350  341 
apply (rule group.l_repr_independence [OF a_group, 
20318
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

342 
folded a_l_coset_def, simplified monoid_record_simps]) 
23350  343 
apply (rule y) 
344 
apply (rule x) 

345 
apply (rule sb) 

346 
done 

20318
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

347 

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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

348 
lemma (in abelian_group) setadd_subset_G: 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

349 
"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G\<rbrakk> \<Longrightarrow> H <+> K \<subseteq> carrier G" 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

350 
by (rule group.setmult_subset_G [OF a_group, 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

351 
folded set_add_def, simplified monoid_record_simps]) 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

352 

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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

353 
lemma (in abelian_group) subgroup_add_id: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> \<Longrightarrow> H <+> H = H" 
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Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

354 
by (rule group.subgroup_mult_id [OF a_group, 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

355 
folded set_add_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

356 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
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parents:
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changeset

357 
lemma (in abelian_subgroup) a_rcos_inv: 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

358 
assumes x: "x \<in> carrier G" 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

359 
shows "a_set_inv (H +> x) = H +> (\<ominus> x)" 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

360 
by (rule normal.rcos_inv [OF a_normal, 
23350  361 
folded a_r_coset_def a_inv_def A_SET_INV_def, simplified monoid_record_simps]) (rule x) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

362 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

363 
lemma (in abelian_group) a_setmult_rcos_assoc: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

364 
"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk> 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

365 
\<Longrightarrow> H <+> (K +> x) = (H <+> K) +> x" 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

366 
by (rule group.setmult_rcos_assoc [OF a_group, 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

367 
folded set_add_def a_r_coset_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

368 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

369 
lemma (in abelian_group) a_rcos_assoc_lcos: 
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Restructured algebra library, added ideals and quotient rings.
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parents:
diff
changeset

370 
"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk> 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

371 
\<Longrightarrow> (H +> x) <+> K = H <+> (x <+ K)" 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

372 
by (rule group.rcos_assoc_lcos [OF a_group, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

373 
folded set_add_def a_r_coset_def a_l_coset_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

374 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

375 
lemma (in abelian_subgroup) a_rcos_sum: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

376 
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

377 
\<Longrightarrow> (H +> x) <+> (H +> y) = H +> (x \<oplus> y)" 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

378 
by (rule normal.rcos_sum [OF a_normal, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

379 
folded set_add_def a_r_coset_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

380 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

381 
lemma (in abelian_subgroup) rcosets_add_eq: 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

382 
"M \<in> a_rcosets H \<Longrightarrow> H <+> M = M" 
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

383 
 {* generalizes @{text subgroup_mult_id} *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

384 
by (rule normal.rcosets_mult_eq [OF a_normal, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

385 
folded set_add_def A_RCOSETS_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

386 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

387 

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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

388 
subsection {* Congruence Relation *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

389 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

390 
lemma (in abelian_subgroup) a_equiv_rcong: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

391 
shows "equiv (carrier G) (racong H)" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

392 
by (rule subgroup.equiv_rcong [OF a_subgroup a_group, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

393 
folded a_r_congruent_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

394 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

395 
lemma (in abelian_subgroup) a_l_coset_eq_rcong: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

396 
assumes a: "a \<in> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

397 
shows "a <+ H = racong H `` {a}" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

398 
by (rule subgroup.l_coset_eq_rcong [OF a_subgroup a_group, 
23350  399 
folded a_r_congruent_def a_l_coset_def, simplified monoid_record_simps]) (rule a) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

400 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

401 
lemma (in abelian_subgroup) a_rcos_equation: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

402 
shows 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

403 
"\<lbrakk>ha \<oplus> a = h \<oplus> b; a \<in> carrier G; b \<in> carrier G; 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

404 
h \<in> H; ha \<in> H; hb \<in> H\<rbrakk> 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

405 
\<Longrightarrow> hb \<oplus> a \<in> (\<Union>h\<in>H. {h \<oplus> b})" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

406 
by (rule group.rcos_equation [OF a_group a_subgroup, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

407 
folded a_r_congruent_def a_l_coset_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

408 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

409 
lemma (in abelian_subgroup) a_rcos_disjoint: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

410 
shows "\<lbrakk>a \<in> a_rcosets H; b \<in> a_rcosets H; a\<noteq>b\<rbrakk> \<Longrightarrow> a \<inter> b = {}" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

411 
by (rule group.rcos_disjoint [OF a_group a_subgroup, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

412 
folded A_RCOSETS_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

413 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

414 
lemma (in abelian_subgroup) a_rcos_self: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

415 
shows "x \<in> carrier G \<Longrightarrow> x \<in> H +> x" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

416 
by (rule group.rcos_self [OF a_group a_subgroup, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

417 
folded a_r_coset_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

418 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

419 
lemma (in abelian_subgroup) a_rcosets_part_G: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

420 
shows "\<Union>(a_rcosets H) = carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

421 
by (rule group.rcosets_part_G [OF a_group a_subgroup, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

422 
folded A_RCOSETS_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

423 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

424 
lemma (in abelian_subgroup) a_cosets_finite: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

425 
"\<lbrakk>c \<in> a_rcosets H; H \<subseteq> carrier G; finite (carrier G)\<rbrakk> \<Longrightarrow> finite c" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

426 
by (rule group.cosets_finite [OF a_group, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

427 
folded A_RCOSETS_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

428 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

429 
lemma (in abelian_group) a_card_cosets_equal: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

430 
"\<lbrakk>c \<in> a_rcosets H; H \<subseteq> carrier G; finite(carrier G)\<rbrakk> 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

431 
\<Longrightarrow> card c = card H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

432 
by (rule group.card_cosets_equal [OF a_group, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

433 
folded A_RCOSETS_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

434 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

435 
lemma (in abelian_group) rcosets_subset_PowG: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

436 
"additive_subgroup H G \<Longrightarrow> a_rcosets H \<subseteq> Pow(carrier G)" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

437 
by (rule group.rcosets_subset_PowG [OF a_group, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

438 
folded A_RCOSETS_def, simplified monoid_record_simps], 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

439 
rule additive_subgroup.a_subgroup) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

440 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

441 
theorem (in abelian_group) a_lagrange: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

442 
"\<lbrakk>finite(carrier G); additive_subgroup H G\<rbrakk> 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

443 
\<Longrightarrow> card(a_rcosets H) * card(H) = order(G)" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

444 
by (rule group.lagrange [OF a_group, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

445 
folded A_RCOSETS_def, simplified monoid_record_simps order_def, folded order_def]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

446 
(fast intro!: additive_subgroup.a_subgroup)+ 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

447 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

448 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

449 
subsection {* Factorization *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

450 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

451 
lemmas A_FactGroup_defs = A_FactGroup_def FactGroup_def 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

452 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

453 
lemma A_FactGroup_def': 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

454 
includes struct G 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

455 
shows "G A_Mod H \<equiv> \<lparr>carrier = a_rcosets\<^bsub>G\<^esub> H, mult = set_add G, one = H\<rparr>" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

456 
unfolding A_FactGroup_defs 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

457 
by (fold A_RCOSETS_def set_add_def) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

458 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

459 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

460 
lemma (in abelian_subgroup) a_setmult_closed: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

461 
"\<lbrakk>K1 \<in> a_rcosets H; K2 \<in> a_rcosets H\<rbrakk> \<Longrightarrow> K1 <+> K2 \<in> a_rcosets H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

462 
by (rule normal.setmult_closed [OF a_normal, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

463 
folded A_RCOSETS_def set_add_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

464 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

465 
lemma (in abelian_subgroup) a_setinv_closed: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

466 
"K \<in> a_rcosets H \<Longrightarrow> a_set_inv K \<in> a_rcosets H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

467 
by (rule normal.setinv_closed [OF a_normal, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

468 
folded A_RCOSETS_def A_SET_INV_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

469 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

470 
lemma (in abelian_subgroup) a_rcosets_assoc: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

471 
"\<lbrakk>M1 \<in> a_rcosets H; M2 \<in> a_rcosets H; M3 \<in> a_rcosets H\<rbrakk> 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

472 
\<Longrightarrow> M1 <+> M2 <+> M3 = M1 <+> (M2 <+> M3)" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

473 
by (rule normal.rcosets_assoc [OF a_normal, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

474 
folded A_RCOSETS_def set_add_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

475 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

476 
lemma (in abelian_subgroup) a_subgroup_in_rcosets: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

477 
"H \<in> a_rcosets H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

478 
by (rule subgroup.subgroup_in_rcosets [OF a_subgroup a_group, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

479 
folded A_RCOSETS_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

480 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

481 
lemma (in abelian_subgroup) a_rcosets_inv_mult_group_eq: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

482 
"M \<in> a_rcosets H \<Longrightarrow> a_set_inv M <+> M = H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

483 
by (rule normal.rcosets_inv_mult_group_eq [OF a_normal, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

484 
folded A_RCOSETS_def A_SET_INV_def set_add_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

485 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

486 
theorem (in abelian_subgroup) a_factorgroup_is_group: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

487 
"group (G A_Mod H)" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

488 
by (rule normal.factorgroup_is_group [OF a_normal, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

489 
folded A_FactGroup_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

490 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

491 
text {* Since the Factorization is based on an \emph{abelian} subgroup, is results in 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

492 
a commutative group *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

493 
theorem (in abelian_subgroup) a_factorgroup_is_comm_group: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

494 
"comm_group (G A_Mod H)" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

495 
apply (intro comm_group.intro comm_monoid.intro) prefer 3 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

496 
apply (rule a_factorgroup_is_group) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

497 
apply (rule group.axioms[OF a_factorgroup_is_group]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

498 
apply (rule comm_monoid_axioms.intro) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

499 
apply (unfold A_FactGroup_def FactGroup_def RCOSETS_def, fold set_add_def a_r_coset_def, clarsimp) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

500 
apply (simp add: a_rcos_sum a_comm) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

501 
done 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

502 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

503 
lemma add_A_FactGroup [simp]: "X \<otimes>\<^bsub>(G A_Mod H)\<^esub> X' = X <+>\<^bsub>G\<^esub> X'" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

504 
by (simp add: A_FactGroup_def set_add_def) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

505 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

506 
lemma (in abelian_subgroup) a_inv_FactGroup: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

507 
"X \<in> carrier (G A_Mod H) \<Longrightarrow> inv\<^bsub>G A_Mod H\<^esub> X = a_set_inv X" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

508 
by (rule normal.inv_FactGroup [OF a_normal, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

509 
folded A_FactGroup_def A_SET_INV_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

510 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

511 
text{*The coset map is a homomorphism from @{term G} to the quotient group 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

512 
@{term "G Mod H"}*} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

513 
lemma (in abelian_subgroup) a_r_coset_hom_A_Mod: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

514 
"(\<lambda>a. H +> a) \<in> hom \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> (G A_Mod H)" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

515 
by (rule normal.r_coset_hom_Mod [OF a_normal, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

516 
folded A_FactGroup_def a_r_coset_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

517 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

518 
text {* The isomorphism theorems have been omitted from lifting, at 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

519 
least for now *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

520 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

521 
subsection{*The First Isomorphism Theorem*} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

522 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

523 
text{*The quotient by the kernel of a homomorphism is isomorphic to the 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

524 
range of that homomorphism.*} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

525 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

526 
lemmas a_kernel_defs = 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

527 
a_kernel_def kernel_def 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

528 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

529 
lemma a_kernel_def': 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

530 
"a_kernel R S h \<equiv> {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

531 
by (rule a_kernel_def[unfolded kernel_def, simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

532 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

533 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

534 
subsection {* Homomorphisms *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

535 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

536 
lemma abelian_group_homI: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

537 
includes abelian_group G 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

538 
includes abelian_group H 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

539 
assumes a_group_hom: "group_hom ( carrier = carrier G, mult = add G, one = zero G ) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

540 
( carrier = carrier H, mult = add H, one = zero H ) h" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

541 
shows "abelian_group_hom G H h" 
23350  542 
apply (intro abelian_group_hom.intro abelian_group_hom_axioms.intro) 
543 
apply (rule G.abelian_group_axioms) 

544 
apply (rule H.abelian_group_axioms) 

545 
apply (rule a_group_hom) 

546 
done 

20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

547 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

548 
lemma (in abelian_group_hom) is_abelian_group_hom: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

549 
"abelian_group_hom G H h" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

550 
by (unfold_locales) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

551 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

552 
lemma (in abelian_group_hom) hom_add [simp]: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

553 
"[ x : carrier G; y : carrier G ] 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

554 
==> h (x \<oplus>\<^bsub>G\<^esub> y) = h x \<oplus>\<^bsub>H\<^esub> h y" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

555 
by (rule group_hom.hom_mult[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

556 
simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

557 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

558 
lemma (in abelian_group_hom) hom_closed [simp]: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

559 
"x \<in> carrier G \<Longrightarrow> h x \<in> carrier H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

560 
by (rule group_hom.hom_closed[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

561 
simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

562 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

563 
lemma (in abelian_group_hom) zero_closed [simp]: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

564 
"h \<zero> \<in> carrier H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

565 
by (rule group_hom.one_closed[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

566 
simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

567 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

568 
lemma (in abelian_group_hom) hom_zero [simp]: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

569 
"h \<zero> = \<zero>\<^bsub>H\<^esub>" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

570 
by (rule group_hom.hom_one[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

571 
simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

572 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

573 
lemma (in abelian_group_hom) a_inv_closed [simp]: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

574 
"x \<in> carrier G ==> h (\<ominus>x) \<in> carrier H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

575 
by (rule group_hom.inv_closed[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

576 
folded a_inv_def, simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

577 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

578 
lemma (in abelian_group_hom) hom_a_inv [simp]: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

579 
"x \<in> carrier G ==> h (\<ominus>x) = \<ominus>\<^bsub>H\<^esub> (h x)" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

580 
by (rule group_hom.hom_inv[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

581 
folded a_inv_def, simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

582 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

583 
lemma (in abelian_group_hom) additive_subgroup_a_kernel: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

584 
"additive_subgroup (a_kernel G H h) G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

585 
apply (rule additive_subgroup.intro) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

586 
apply (rule group_hom.subgroup_kernel[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

587 
folded a_kernel_def, simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

588 
done 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

589 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

590 
text{*The kernel of a homomorphism is an abelian subgroup*} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

591 
lemma (in abelian_group_hom) abelian_subgroup_a_kernel: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

592 
"abelian_subgroup (a_kernel G H h) G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

593 
apply (rule abelian_subgroupI) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

594 
apply (rule group_hom.normal_kernel[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

595 
folded a_kernel_def, simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

596 
apply (simp add: G.a_comm) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

597 
done 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

598 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

599 
lemma (in abelian_group_hom) A_FactGroup_nonempty: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

600 
assumes X: "X \<in> carrier (G A_Mod a_kernel G H h)" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

601 
shows "X \<noteq> {}" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

602 
by (rule group_hom.FactGroup_nonempty[OF a_group_hom, 
23350  603 
folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule X) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

604 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

605 
lemma (in abelian_group_hom) FactGroup_contents_mem: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

606 
assumes X: "X \<in> carrier (G A_Mod (a_kernel G H h))" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

607 
shows "contents (h`X) \<in> carrier H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

608 
by (rule group_hom.FactGroup_contents_mem[OF a_group_hom, 
23350  609 
folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule X) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

610 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

611 
lemma (in abelian_group_hom) A_FactGroup_hom: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

612 
"(\<lambda>X. contents (h`X)) \<in> hom (G A_Mod (a_kernel G H h)) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

613 
\<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr>" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

614 
by (rule group_hom.FactGroup_hom[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

615 
folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

616 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

617 
lemma (in abelian_group_hom) A_FactGroup_inj_on: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

618 
"inj_on (\<lambda>X. contents (h ` X)) (carrier (G A_Mod a_kernel G H h))" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

619 
by (rule group_hom.FactGroup_inj_on[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

620 
folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

621 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

622 
text{*If the homomorphism @{term h} is onto @{term H}, then so is the 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

623 
homomorphism from the quotient group*} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

624 
lemma (in abelian_group_hom) A_FactGroup_onto: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

625 
assumes h: "h ` carrier G = carrier H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

626 
shows "(\<lambda>X. contents (h ` X)) ` carrier (G A_Mod a_kernel G H h) = carrier H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

627 
by (rule group_hom.FactGroup_onto[OF a_group_hom, 
23350  628 
folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule h) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

629 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

630 
text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

631 
quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

632 
theorem (in abelian_group_hom) A_FactGroup_iso: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

633 
"h ` carrier G = carrier H 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

634 
\<Longrightarrow> (\<lambda>X. contents (h`X)) \<in> (G A_Mod (a_kernel G H h)) \<cong> 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

635 
( carrier = carrier H, mult = add H, one = zero H )" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

636 
by (rule group_hom.FactGroup_iso[OF a_group_hom, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

637 
folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

638 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

639 
section {* Lemmas Lifted from CosetExt.thy *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

640 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

641 
text {* Not eveything from \texttt{CosetExt.thy} is lifted here. *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

642 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

643 
subsection {* General Lemmas from \texttt{AlgebraExt.thy} *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

644 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

645 
lemma (in additive_subgroup) a_Hcarr [simp]: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

646 
assumes hH: "h \<in> H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

647 
shows "h \<in> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

648 
by (rule subgroup.mem_carrier [OF a_subgroup, 
23350  649 
simplified monoid_record_simps]) (rule hH) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

650 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

651 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

652 
subsection {* Lemmas for Right Cosets *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

653 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

654 
lemma (in abelian_subgroup) a_elemrcos_carrier: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

655 
assumes acarr: "a \<in> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

656 
and a': "a' \<in> H +> a" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

657 
shows "a' \<in> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

658 
by (rule subgroup.elemrcos_carrier [OF a_subgroup a_group, 
23350  659 
folded a_r_coset_def, simplified monoid_record_simps]) (rule acarr, rule a') 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

660 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

661 
lemma (in abelian_subgroup) a_rcos_const: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

662 
assumes hH: "h \<in> H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

663 
shows "H +> h = H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

664 
by (rule subgroup.rcos_const [OF a_subgroup a_group, 
23350  665 
folded a_r_coset_def, simplified monoid_record_simps]) (rule hH) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

666 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

667 
lemma (in abelian_subgroup) a_rcos_module_imp: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

668 
assumes xcarr: "x \<in> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

669 
and x'cos: "x' \<in> H +> x" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

670 
shows "(x' \<oplus> \<ominus>x) \<in> H" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

671 
by (rule subgroup.rcos_module_imp [OF a_subgroup a_group, 
23350  672 
folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) (rule xcarr, rule x'cos) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

673 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

674 
lemma (in abelian_subgroup) a_rcos_module_rev: 
23350  675 
assumes "x \<in> carrier G" "x' \<in> carrier G" 
676 
and "(x' \<oplus> \<ominus>x) \<in> H" 

20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

677 
shows "x' \<in> H +> x" 
23350  678 
using assms 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

679 
by (rule subgroup.rcos_module_rev [OF a_subgroup a_group, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

680 
folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

681 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

682 
lemma (in abelian_subgroup) a_rcos_module: 
23350  683 
assumes "x \<in> carrier G" "x' \<in> carrier G" 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

684 
shows "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> H)" 
23350  685 
using assms 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

686 
by (rule subgroup.rcos_module [OF a_subgroup a_group, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

687 
folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

688 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

689 
"variant" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

690 
lemma (in abelian_subgroup) a_rcos_module_minus: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

691 
includes ring G 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

692 
assumes carr: "x \<in> carrier G" "x' \<in> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

693 
shows "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

694 
proof  
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

695 
from carr 
23350  696 
have "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> H)" by (rule a_rcos_module) 
697 
with carr 

698 
show "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)" 

699 
by (simp add: minus_eq) 

20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

700 
qed 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

701 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

702 
lemma (in abelian_subgroup) a_repr_independence': 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

703 
assumes "y \<in> H +> x" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

704 
and "x \<in> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

705 
shows "H +> x = H +> y" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

706 
apply (rule a_repr_independence, assumption+) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

707 
apply (rule a_subgroup) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

708 
done 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

709 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

710 
lemma (in abelian_subgroup) a_repr_independenceD: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

711 
assumes ycarr: "y \<in> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

712 
and repr: "H +> x = H +> y" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

713 
shows "y \<in> H +> x" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

714 
by (rule group.repr_independenceD [OF a_group a_subgroup, 
23383  715 
folded a_r_coset_def, simplified monoid_record_simps]) (rule ycarr, rule repr) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

716 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

717 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

718 
subsection {* Lemmas for the Set of Right Cosets *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

719 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

720 
lemma (in abelian_subgroup) a_rcosets_carrier: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

721 
"X \<in> a_rcosets H \<Longrightarrow> X \<subseteq> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

722 
by (rule subgroup.rcosets_carrier [OF a_subgroup a_group, 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

723 
folded A_RCOSETS_def, simplified monoid_record_simps]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

724 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

725 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

726 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

727 
subsection {* Addition of Subgroups *} 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

728 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

729 
lemma (in abelian_monoid) set_add_closed: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

730 
assumes Acarr: "A \<subseteq> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

731 
and Bcarr: "B \<subseteq> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

732 
shows "A <+> B \<subseteq> carrier G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

733 
by (rule monoid.set_mult_closed [OF a_monoid, 
23383  734 
folded set_add_def, simplified monoid_record_simps]) (rule Acarr, rule Bcarr) 
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

735 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

736 
lemma (in abelian_group) add_additive_subgroups: 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

737 
assumes subH: "additive_subgroup H G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

738 
and subK: "additive_subgroup K G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

739 
shows "additive_subgroup (H <+> K) G" 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

740 
apply (rule additive_subgroup.intro) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

741 
apply (unfold set_add_def) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

742 
apply (intro comm_group.mult_subgroups) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

743 
apply (rule a_comm_group) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

744 
apply (rule additive_subgroup.a_subgroup[OF subH]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

745 
apply (rule additive_subgroup.a_subgroup[OF subK]) 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

746 
done 
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

747 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
diff
changeset

748 
end 