(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar (ZF logic). Copyright (C) 2005, 2006 Slawomir Kolodynski This program is free software; Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Groups 2› theory Group_ZF_2 imports AbelianGroup_ZF func_ZF EquivClass1 begin text{*This theory continues Group\_ZF.thy and considers lifting the group structure to function spaces and projecting the group structure to quotient spaces, in particular the quotient qroup.*} subsection{*Lifting groups to function spaces*} text{*If we have a monoid (group) $G$ than we get a monoid (group) structure on a space of functions valued in in $G$ by defining $(f\cdot g)(x) := f(x)\cdot g(x)$. We call this process ''lifting the monoid (group) to function space''. This section formalizes this lifting.*} text{*The lifted operation is an operation on the function space.*} lemma (in monoid0) Group_ZF_2_1_L0A: assumes A1: "F = f {lifted to function space over} X" shows "F : (X→G)×(X→G)→(X→G)" proof - from monoidAsssum have "f : G×G→G" using IsAmonoid_def IsAssociative_def by simp with A1 show ?thesis using func_ZF_1_L3 group0_1_L3B by auto qed text{*The result of the lifted operation is in the function space.*} lemma (in monoid0) Group_ZF_2_1_L0: assumes A1:"F = f {lifted to function space over} X" and A2:"s:X→G" "r:X→G" shows "F`⟨ s,r⟩ : X→G" proof - from A1 have "F : (X→G)×(X→G)→(X→G)" using Group_ZF_2_1_L0A by simp with A2 show ?thesis using apply_funtype by simp qed text{*The lifted monoid operation has a neutral element, namely the constant function with the neutral element as the value. *} lemma (in monoid0) Group_ZF_2_1_L1: assumes A1: "F = f {lifted to function space over} X" and A2: "E = ConstantFunction(X,TheNeutralElement(G,f))" shows "E : X→G ∧ (∀s∈X→G. F`⟨ E,s⟩ = s ∧ F`⟨ s,E⟩ = s)" proof from A2 show T1:"E : X→G" using unit_is_neutral func1_3_L1 by simp show "∀s∈X→G. F`⟨ E,s⟩ = s ∧ F`⟨ s,E⟩ = s" proof fix s assume A3:"s:X→G" from monoidAsssum have T2:"f : G×G→G" using IsAmonoid_def IsAssociative_def by simp from A3 A1 T1 have "F`⟨ E,s⟩ : X→G" "F`⟨ s,E⟩ : X→G" "s : X→G" using Group_ZF_2_1_L0 by auto moreover from T2 A1 T1 A2 A3 have "∀x∈X. (F`⟨ E,s⟩)`(x) = s`(x)" "∀x∈X. (F`⟨ s,E⟩)`(x) = s`(x)" using func_ZF_1_L4 group0_1_L3B func1_3_L2 apply_type unit_is_neutral by auto ultimately show "F`⟨ E,s⟩ = s ∧ F`⟨ s,E⟩ = s" using fun_extension_iff by auto qed qed text{*Monoids can be lifted to a function space.*} lemma (in monoid0) Group_ZF_2_1_T1: assumes A1: "F = f {lifted to function space over} X" shows "IsAmonoid(X→G,F)" proof - from monoidAsssum A1 have "F {is associative on} (X→G)" using IsAmonoid_def func_ZF_2_L4 group0_1_L3B by auto moreover from A1 have "∃ E ∈ X→G. ∀s ∈ X→G. F`⟨ E,s⟩ = s ∧ F`⟨ s,E⟩ = s" using Group_ZF_2_1_L1 by blast ultimately show ?thesis using IsAmonoid_def by simp qed text{*The constant function with the neutral element as the value is the neutral element of the lifted monoid.*} lemma Group_ZF_2_1_L2: assumes A1: "IsAmonoid(G,f)" and A2: "F = f {lifted to function space over} X" and A3: "E = ConstantFunction(X,TheNeutralElement(G,f))" shows "E = TheNeutralElement(X→G,F)" proof - from A1 A2 have T1:"monoid0(G,f)" and T2:"monoid0(X→G,F)" using monoid0_def monoid0.Group_ZF_2_1_T1 by auto from T1 A2 A3 have "E : X→G ∧ (∀s∈X→G. F`⟨ E,s⟩ = s ∧ F`⟨ s,E⟩ = s)" using monoid0.Group_ZF_2_1_L1 by simp with T2 show ?thesis using monoid0.group0_1_L4 by auto qed text{*The lifted operation acts on the functions in a natural way defined by the monoid operation.*} lemma (in monoid0) lifted_val: assumes "F = f {lifted to function space over} X" and "s:X→G" "r:X→G" and "x∈X" shows "(F`⟨s,r⟩)`(x) = s`(x) ⊕ r`(x)" using monoidAsssum assms IsAmonoid_def IsAssociative_def group0_1_L3B func_ZF_1_L4 by auto text{*The lifted operation acts on the functions in a natural way defined by the group operation. This is the same as @{text "lifted_val"}, but in the @{text "group0"} context.*} lemma (in group0) Group_ZF_2_1_L3: assumes "F = P {lifted to function space over} X" and "s:X→G" "r:X→G" and "x∈X" shows "(F`⟨s,r⟩)`(x) = s`(x)⋅r`(x)" using assms group0_2_L1 monoid0.lifted_val by simp text{*In the group0 context we can apply theorems proven in monoid0 context to the lifted monoid.*} lemma (in group0) Group_ZF_2_1_L4: assumes A1: "F = P {lifted to function space over} X" shows "monoid0(X→G,F)" proof - from A1 show ?thesis using group0_2_L1 monoid0.Group_ZF_2_1_T1 monoid0_def by simp qed text{*The compostion of a function $f:X\rightarrow G$ with the group inverse is a right inverse for the lifted group. *} lemma (in group0) Group_ZF_2_1_L5: assumes A1: "F = P {lifted to function space over} X" and A2: "s : X→G" and A3: "i = GroupInv(G,P) O s" shows "i: X→G" and "F`⟨ s,i⟩ = TheNeutralElement(X→G,F)" proof - let ?E = "ConstantFunction(X,𝟭)" have "?E : X→G" using group0_2_L2 func1_3_L1 by simp moreover from groupAssum A2 A3 A1 have "F`⟨ s,i⟩ : X→G" using group0_2_T2 comp_fun Group_ZF_2_1_L4 monoid0.group0_1_L1 by simp moreover from groupAssum A2 A3 A1 have "∀x∈X. (F`⟨ s,i⟩)`(x) = ?E`(x)" using group0_2_T2 comp_fun Group_ZF_2_1_L3 comp_fun_apply apply_funtype group0_2_L6 func1_3_L2 by simp moreover from groupAssum A1 have "?E = TheNeutralElement(X→G,F)" using IsAgroup_def Group_ZF_2_1_L2 by simp ultimately show "F`⟨ s,i⟩ = TheNeutralElement(X→G,F)" using fun_extension_iff IsAgroup_def Group_ZF_2_1_L2 by simp from groupAssum A2 A3 show "i: X→G" using group0_2_T2 comp_fun by simp qed text{*Groups can be lifted to the function space.*} theorem (in group0) Group_ZF_2_1_T2: assumes A1: "F = P {lifted to function space over} X" shows "IsAgroup(X→G,F)" proof - from A1 have "IsAmonoid(X→G,F)" using group0_2_L1 monoid0.Group_ZF_2_1_T1 by simp moreover have "∀s∈X→G. ∃i∈X→G. F`⟨ s,i⟩ = TheNeutralElement(X→G,F)" proof fix s assume A2: "s : X→G" let ?i = "GroupInv(G,P) O s" from groupAssum A2 have "?i:X→G" using group0_2_T2 comp_fun by simp moreover from A1 A2 have "F`⟨ s,?i⟩ = TheNeutralElement(X→G,F)" using Group_ZF_2_1_L5 by fast ultimately show "∃i∈X→G. F`⟨ s,i⟩ = TheNeutralElement(X→G,F)" by auto qed ultimately show ?thesis using IsAgroup_def by simp qed text{*What is the group inverse for the lifted group?*} lemma (in group0) Group_ZF_2_1_L6: assumes A1: "F = P {lifted to function space over} X" shows "∀s∈(X→G). GroupInv(X→G,F)`(s) = GroupInv(G,P) O s" proof - from A1 have "group0(X→G,F)" using group0_def Group_ZF_2_1_T2 by simp moreover from A1 have "∀s∈X→G. GroupInv(G,P) O s : X→G ∧ F`⟨ s,GroupInv(G,P) O s⟩ = TheNeutralElement(X→G,F)" using Group_ZF_2_1_L5 by simp ultimately have "∀s∈(X→G). GroupInv(G,P) O s = GroupInv(X→G,F)`(s)" by (rule group0.group0_2_L9A) thus ?thesis by simp qed text{*What is the value of the group inverse for the lifted group?*} corollary (in group0) lift_gr_inv_val: assumes "F = P {lifted to function space over} X" and "s : X→G" and "x∈X" shows "(GroupInv(X→G,F)`(s))`(x) = (s`(x))¯" using groupAssum assms Group_ZF_2_1_L6 group0_2_T2 comp_fun_apply by simp text{*What is the group inverse in a subgroup of the lifted group?*} lemma (in group0) Group_ZF_2_1_L6A: assumes A1: "F = P {lifted to function space over} X" and A2: "IsAsubgroup(H,F)" and A3: "g = restrict(F,H×H)" and A4: "s∈H" shows "GroupInv(H,g)`(s) = GroupInv(G,P) O s" proof - from A1 have T1: "group0(X→G,F)" using group0_def Group_ZF_2_1_T2 by simp with A2 A3 A4 have "GroupInv(H,g)`(s) = GroupInv(X→G,F)`(s)" using group0.group0_3_T1 restrict by simp moreover from T1 A1 A2 A4 have "GroupInv(X→G,F)`(s) = GroupInv(G,P) O s" using group0.group0_3_L2 Group_ZF_2_1_L6 by blast ultimately show ?thesis by simp qed text{*If a group is abelian, then its lift to a function space is also abelian.*} lemma (in group0) Group_ZF_2_1_L7: assumes A1: "F = P {lifted to function space over} X" and A2: "P {is commutative on} G" shows "F {is commutative on} (X→G)" proof- from A1 A2 have "F {is commutative on} (X→range(P))" using group_oper_assocA func_ZF_2_L2 by simp moreover from groupAssum have "range(P) = G" using group0_2_L1 monoid0.group0_1_L3B by simp ultimately show ?thesis by simp qed subsection{*Equivalence relations on groups*} text{*The goal of this section is to establish that (under some conditions) given an equivalence relation on a group or (monoid )we can project the group (monoid) structure on the quotient and obtain another group.*} text{*The neutral element class is neutral in the projection.*} lemma (in monoid0) Group_ZF_2_2_L1: assumes A1: "equiv(G,r)" and A2:"Congruent2(r,f)" and A3: "F = ProjFun2(G,r,f)" and A4: "e = TheNeutralElement(G,f)" shows "r``{e} ∈ G//r ∧ (∀c ∈ G//r. F`⟨ r``{e},c⟩ = c ∧ F`⟨ c,r``{e}⟩ = c)" proof from A4 show T1:"r``{e} ∈ G//r" using unit_is_neutral quotientI by simp show "∀c ∈ G//r. F`⟨ r``{e},c⟩ = c ∧ F`⟨ c,r``{e}⟩ = c" proof fix c assume A5:"c ∈ G//r" then obtain g where D1:"g∈G" "c = r``{g}" using quotient_def by auto with A1 A2 A3 A4 D1 show "F`⟨ r``{e},c⟩ = c ∧ F`⟨ c,r``{e}⟩ = c" using unit_is_neutral EquivClass_1_L10 (*group0_1_L3*) by simp qed qed text{*The projected structure is a monoid.*} theorem (in monoid0) Group_ZF_2_2_T1: assumes A1: "equiv(G,r)" and A2: "Congruent2(r,f)" and A3: "F = ProjFun2(G,r,f)" shows "IsAmonoid(G//r,F)" proof - let ?E = "r``{TheNeutralElement(G,f)}" from A1 A2 A3 have "?E ∈ G//r ∧ (∀c∈G//r. F`⟨ ?E,c⟩ = c ∧ F`⟨ c,?E⟩ = c)" using Group_ZF_2_2_L1 by simp hence "∃E∈G//r. ∀ c∈G//r. F`⟨ E,c⟩ = c ∧ F`⟨ c,E⟩ = c" by auto with monoidAsssum A1 A2 A3 show ?thesis using IsAmonoid_def EquivClass_2_T2 by simp qed text{*The class of the neutral element is the neutral element of the projected monoid.*} lemma Group_ZF_2_2_L1: assumes A1: "IsAmonoid(G,f)" and A2: "equiv(G,r)" and A3: "Congruent2(r,f)" and A4: "F = ProjFun2(G,r,f)" and A5: "e = TheNeutralElement(G,f)" shows " r``{e} = TheNeutralElement(G//r,F)" proof - from A1 A2 A3 A4 have T1:"monoid0(G,f)" and T2:"monoid0(G//r,F)" using monoid0_def monoid0.Group_ZF_2_2_T1 by auto from T1 A2 A3 A4 A5 have "r``{e} ∈ G//r ∧ (∀c ∈ G//r. F`⟨ r``{e},c⟩ = c ∧ F`⟨ c,r``{e}⟩ = c)" using monoid0.Group_ZF_2_2_L1 by simp with T2 show ?thesis using monoid0.group0_1_L4 by auto qed text{*The projected operation can be defined in terms of the group operation on representants in a natural way.*} lemma (in group0) Group_ZF_2_2_L2: assumes A1: "equiv(G,r)" and A2: "Congruent2(r,P)" and A3: "F = ProjFun2(G,r,P)" and A4: "a∈G" "b∈G" shows "F`⟨ r``{a},r``{b}⟩ = r``{a⋅b}" proof - from A1 A2 A3 A4 show ?thesis using EquivClass_1_L10 by simp qed text{*The class of the inverse is a right inverse of the class.*} lemma (in group0) Group_ZF_2_2_L3: assumes A1: "equiv(G,r)" and A2: "Congruent2(r,P)" and A3: "F = ProjFun2(G,r,P)" and A4: "a∈G" shows "F`⟨r``{a},r``{a¯}⟩ = TheNeutralElement(G//r,F)" proof - from A1 A2 A3 A4 have "F`⟨r``{a},r``{a¯}⟩ = r``{𝟭}" using inverse_in_group Group_ZF_2_2_L2 group0_2_L6 by simp with groupAssum A1 A2 A3 show ?thesis using IsAgroup_def Group_ZF_2_2_L1 by simp qed text{*The group structure can be projected to the quotient space.*} theorem (in group0) Group_ZF_3_T2: assumes A1: "equiv(G,r)" and A2: "Congruent2(r,P)" shows "IsAgroup(G//r,ProjFun2(G,r,P))" proof - let ?F = "ProjFun2(G,r,P)" let ?E = "TheNeutralElement(G//r,?F)" from groupAssum A1 A2 have "IsAmonoid(G//r,?F)" using IsAgroup_def monoid0_def monoid0.Group_ZF_2_2_T1 by simp moreover have "∀c∈G//r. ∃b∈G//r. ?F`⟨ c,b⟩ = ?E" proof fix c assume A3: "c ∈ G//r" then obtain g where D1: "g∈G" "c = r``{g}" using quotient_def by auto let ?b = "r``{g¯}" from D1 have "?b ∈ G//r" using inverse_in_group quotientI by simp moreover from A1 A2 D1 have "?F`⟨ c,?b⟩ = ?E" using Group_ZF_2_2_L3 by simp ultimately show "∃b∈G//r. ?F`⟨ c,b⟩ = ?E" by auto qed ultimately show ?thesis using IsAgroup_def by simp qed text{*The group inverse (in the projected group) of a class is the class of the inverse.*} lemma (in group0) Group_ZF_2_2_L4: assumes A1: "equiv(G,r)" and A2: "Congruent2(r,P)" and A3: "F = ProjFun2(G,r,P)" and A4: "a∈G" shows "r``{a¯} = GroupInv(G//r,F)`(r``{a})" proof - from A1 A2 A3 have "group0(G//r,F)" using Group_ZF_3_T2 group0_def by simp moreover from A4 have "r``{a} ∈ G//r" "r``{a¯} ∈ G//r" using inverse_in_group quotientI by auto moreover from A1 A2 A3 A4 have "F`⟨r``{a},r``{a¯}⟩ = TheNeutralElement(G//r,F)" using Group_ZF_2_2_L3 by simp ultimately show ?thesis by (rule group0.group0_2_L9) qed subsection{*Normal subgroups and quotient groups*} text{*If $H$ is a subgroup of $G$, then for every $a\in G$ we can cosider the sets $\{a\cdot h. h \in H\}$ and $\{ h\cdot a. h \in H\}$ (called a left and right ''coset of H'', resp.) These sets sometimes form a group, called the ''quotient group''. This section discusses the notion of quotient groups.*} text{*A normal subgorup $N$ of a group $G$ is such that $aba^{-1}$ belongs to $N$ if $a\in G, b\in N$. *} definition "IsAnormalSubgroup(G,P,N) ≡ IsAsubgroup(N,P) ∧ (∀n∈N.∀g∈G. P`⟨ P`⟨ g,n ⟩,GroupInv(G,P)`(g) ⟩ ∈ N)" text{*Having a group and a normal subgroup $N$ we can create another group consisting of eqivalence classes of the relation $a\sim b \equiv a\cdot b^{-1} \in N$. We will refer to this relation as the quotient group relation. The classes of this relation are in fact cosets of subgroup $H$.*} definition "QuotientGroupRel(G,P,H) ≡ {⟨ a,b⟩ ∈ G×G. P`⟨ a, GroupInv(G,P)`(b)⟩ ∈ H}" text{*Next we define the operation in the quotient group as the projection of the group operation on the classses of the quotient group relation.*} definition "QuotientGroupOp(G,P,H) ≡ ProjFun2(G,QuotientGroupRel(G,P,H ),P)" text{*Definition of a normal subgroup in a more readable notation.*} lemma (in group0) Group_ZF_2_4_L0: assumes "IsAnormalSubgroup(G,P,H)" and "g∈G" "n∈H" shows "g⋅n⋅g¯ ∈ H" using assms IsAnormalSubgroup_def by simp text{*The quotient group relation is reflexive.*} lemma (in group0) Group_ZF_2_4_L1: assumes "IsAsubgroup(H,P)" shows "refl(G,QuotientGroupRel(G,P,H))" using assms group0_2_L6 group0_3_L5 QuotientGroupRel_def refl_def by simp text{*The quotient group relation is symmetric.*} lemma (in group0) Group_ZF_2_4_L2: assumes A1:"IsAsubgroup(H,P)" shows "sym(QuotientGroupRel(G,P,H))" proof - { fix a b assume A2: "⟨ a,b⟩ ∈ QuotientGroupRel(G,P,H)" with A1 have "(a⋅b¯)¯ ∈ H" using QuotientGroupRel_def group0_3_T3A by simp moreover from A2 have "(a⋅b¯)¯ = b⋅a¯" using QuotientGroupRel_def group0_2_L12 by simp ultimately have "b⋅a¯ ∈ H" by simp with A2 have "⟨ b,a⟩ ∈ QuotientGroupRel(G,P,H)" using QuotientGroupRel_def by simp } then show ?thesis using symI by simp qed text{*The quotient group relation is transistive.*} lemma (in group0) Group_ZF_2_4_L3A: assumes A1: "IsAsubgroup(H,P)" and A2: "⟨ a,b⟩ ∈ QuotientGroupRel(G,P,H)" and A3: "⟨ b,c⟩ ∈ QuotientGroupRel(G,P,H)" shows "⟨ a,c⟩ ∈ QuotientGroupRel(G,P,H)" proof - let ?r = "QuotientGroupRel(G,P,H)" from A2 A3 have T1:"a∈G" "b∈G" "c∈G" using QuotientGroupRel_def by auto from A1 A2 A3 have "(a⋅b¯)⋅(b⋅c¯) ∈ H" using QuotientGroupRel_def group0_3_L6 by simp moreover from T1 have "a⋅c¯ = (a⋅b¯)⋅(b⋅c¯)" using group0_2_L14A by blast ultimately have "a⋅c¯ ∈ H" by simp with T1 show ?thesis using QuotientGroupRel_def by simp qed text{*The quotient group relation is an equivalence relation. Note we do not need the subgroup to be normal for this to be true.*} lemma (in group0) Group_ZF_2_4_L3: assumes A1:"IsAsubgroup(H,P)" shows "equiv(G,QuotientGroupRel(G,P,H))" proof - let ?r = "QuotientGroupRel(G,P,H)" from A1 have "∀a b c. (⟨a, b⟩ ∈ ?r ∧ ⟨b, c⟩ ∈ ?r ⟶ ⟨a, c⟩ ∈ ?r)" using Group_ZF_2_4_L3A by blast then have "trans(?r)" using Fol1_L2 by blast with A1 show ?thesis using Group_ZF_2_4_L1 Group_ZF_2_4_L2 QuotientGroupRel_def equiv_def by auto qed text{*The next lemma states the essential condition for congruency of the group operation with respect to the quotient group relation.*} lemma (in group0) Group_ZF_2_4_L4: assumes A1: "IsAnormalSubgroup(G,P,H)" and A2: "⟨a1,a2⟩ ∈ QuotientGroupRel(G,P,H)" and A3: "⟨b1,b2⟩ ∈ QuotientGroupRel(G,P,H)" shows "⟨a1⋅b1, a2⋅b2⟩ ∈ QuotientGroupRel(G,P,H)" proof - from A2 A3 have T1: "a1∈G" "a2∈G" "b1∈G" "b2∈G" "a1⋅b1 ∈ G" "a2⋅b2 ∈ G" "b1⋅b2¯ ∈ H" "a1⋅a2¯ ∈ H" using QuotientGroupRel_def group0_2_L1 monoid0.group0_1_L1 by auto with A1 show ?thesis using IsAnormalSubgroup_def group0_3_L6 group0_2_L15 QuotientGroupRel_def by simp qed text{*If the subgroup is normal, the group operation is congruent with respect to the quotient group relation.*} lemma Group_ZF_2_4_L5A: assumes "IsAgroup(G,P)" and "IsAnormalSubgroup(G,P,H)" shows "Congruent2(QuotientGroupRel(G,P,H),P)" using assms group0_def group0.Group_ZF_2_4_L4 Congruent2_def by simp text{*The quotient group is indeed a group.*} theorem Group_ZF_2_4_T1: assumes "IsAgroup(G,P)" and "IsAnormalSubgroup(G,P,H)" shows "IsAgroup(G//QuotientGroupRel(G,P,H),QuotientGroupOp(G,P,H))" using assms group0_def group0.Group_ZF_2_4_L3 IsAnormalSubgroup_def Group_ZF_2_4_L5A group0.Group_ZF_3_T2 QuotientGroupOp_def by simp text{*The class (coset) of the neutral element is the neutral element of the quotient group.*} lemma Group_ZF_2_4_L5B: assumes "IsAgroup(G,P)" and "IsAnormalSubgroup(G,P,H)" and "r = QuotientGroupRel(G,P,H)" and "e = TheNeutralElement(G,P)" shows " r``{e} = TheNeutralElement(G//r,QuotientGroupOp(G,P,H))" using assms IsAnormalSubgroup_def group0_def IsAgroup_def group0.Group_ZF_2_4_L3 Group_ZF_2_4_L5A QuotientGroupOp_def Group_ZF_2_2_L1 by simp text{*A group element is equivalent to the neutral element iff it is in the subgroup we divide the group by.*} lemma (in group0) Group_ZF_2_4_L5C: assumes "a∈G" shows "⟨a,𝟭⟩ ∈ QuotientGroupRel(G,P,H) ⟷ a∈H" using assms QuotientGroupRel_def group_inv_of_one group0_2_L2 by auto text{*A group element is in $H$ iff its class is the neutral element of $G/H$.*} lemma (in group0) Group_ZF_2_4_L5D: assumes A1: "IsAnormalSubgroup(G,P,H)" and A2: "a∈G" and A3: "r = QuotientGroupRel(G,P,H)" and A4: "TheNeutralElement(G//r,QuotientGroupOp(G,P,H)) = e" shows "r``{a} = e ⟷ ⟨a,𝟭⟩ ∈ r" proof assume "r``{a} = e" with groupAssum assms have "r``{𝟭} = r``{a}" and I: "equiv(G,r)" using Group_ZF_2_4_L5B IsAnormalSubgroup_def Group_ZF_2_4_L3 by auto with A2 have "⟨𝟭,a⟩ ∈ r" using eq_equiv_class by simp with I show "⟨a,𝟭⟩ ∈ r" by (rule equiv_is_sym) next assume "⟨a,𝟭⟩ ∈ r" moreover from A1 A3 have "equiv(G,r)" using IsAnormalSubgroup_def Group_ZF_2_4_L3 by simp ultimately have "r``{a} = r``{𝟭}" using equiv_class_eq by simp with groupAssum A1 A3 A4 show "r``{a} = e" using Group_ZF_2_4_L5B by simp qed text{*The class of $a\in G$ is the neutral element of the quotient $G/H$ iff $a\in H$.*} lemma (in group0) Group_ZF_2_4_L5E: assumes "IsAnormalSubgroup(G,P,H)" and "a∈G" and "r = QuotientGroupRel(G,P,H)" and "TheNeutralElement(G//r,QuotientGroupOp(G,P,H)) = e" shows "r``{a} = e ⟷ a∈H" using assms Group_ZF_2_4_L5C Group_ZF_2_4_L5D by simp text{*Essential condition to show that every subgroup of an abelian group is normal.*} lemma (in group0) Group_ZF_2_4_L5: assumes A1: "P {is commutative on} G" and A2: "IsAsubgroup(H,P)" and A3: "g∈G" "h∈H" shows "g⋅h⋅g¯ ∈ H" proof - from A2 A3 have T1:"h∈G" "g¯ ∈ G" using group0_3_L2 inverse_in_group by auto with A3 A1 have "g⋅h⋅g¯ = g¯⋅g⋅h" using group0_4_L4A by simp with A3 T1 show ?thesis using group0_2_L6 group0_2_L2 by simp qed text{*Every subgroup of an abelian group is normal. Moreover, the quotient group is also abelian.*} lemma Group_ZF_2_4_L6: assumes A1: "IsAgroup(G,P)" and A2: "P {is commutative on} G" and A3: "IsAsubgroup(H,P)" shows "IsAnormalSubgroup(G,P,H)" "QuotientGroupOp(G,P,H) {is commutative on} (G//QuotientGroupRel(G,P,H))" proof - from A1 A2 A3 show T1: "IsAnormalSubgroup(G,P,H)" using group0_def IsAnormalSubgroup_def group0.Group_ZF_2_4_L5 by simp let ?r = "QuotientGroupRel(G,P,H)" from A1 A3 T1 have "equiv(G,?r)" "Congruent2(?r,P)" using group0_def group0.Group_ZF_2_4_L3 Group_ZF_2_4_L5A by auto with A2 show "QuotientGroupOp(G,P,H) {is commutative on} (G//QuotientGroupRel(G,P,H))" using EquivClass_2_T1 QuotientGroupOp_def by simp qed text{*The group inverse (in the quotient group) of a class (coset) is the class of the inverse.*} lemma (in group0) Group_ZF_2_4_L7: assumes "IsAnormalSubgroup(G,P,H)" and "a∈G" and "r = QuotientGroupRel(G,P,H)" and "F = QuotientGroupOp(G,P,H)" shows "r``{a¯} = GroupInv(G//r,F)`(r``{a})" using groupAssum assms IsAnormalSubgroup_def Group_ZF_2_4_L3 Group_ZF_2_4_L5A QuotientGroupOp_def Group_ZF_2_2_L4 by simp subsection{*Function spaces as monoids*} text{*On every space of functions $\{f : X\rightarrow X\}$ we can define a natural monoid structure with composition as the operation. This section explores this fact.*} text{*The next lemma states that composition has a neutral element, namely the identity function on $X$ (the one that maps $x\in X$ into itself).*} lemma Group_ZF_2_5_L1: assumes A1: "F = Composition(X)" shows "∃I∈(X→X). ∀f∈(X→X). F`⟨ I,f⟩ = f ∧ F`⟨ f,I⟩ = f" proof- let ?I = "id(X)" from A1 have "?I ∈ X→X ∧ (∀f∈(X→X). F`⟨ ?I,f⟩ = f ∧ F`⟨ f,?I⟩ = f)" using id_type func_ZF_6_L1A by simp thus ?thesis by auto qed text{*The space of functions that map a set $X$ into itsef is a monoid with composition as operation and the identity function as the neutral element.*} lemma Group_ZF_2_5_L2: shows "IsAmonoid(X→X,Composition(X))" "id(X) = TheNeutralElement(X→X,Composition(X))" proof - let ?I = "id(X)" let ?F = "Composition(X)" show "IsAmonoid(X→X,Composition(X))" using func_ZF_5_L5 Group_ZF_2_5_L1 IsAmonoid_def by auto then have "monoid0(X→X,?F)" using monoid0_def by simp moreover have "?I ∈ X→X ∧ (∀f∈(X→X). ?F`⟨ ?I,f⟩ = f ∧ ?F`⟨ f,?I⟩ = f)" using id_type func_ZF_6_L1A by simp ultimately show "?I = TheNeutralElement(X→X,?F)" using monoid0.group0_1_L4 by auto qed end