(*

This file is a part of IsarMathLib -

a library of formalized mathematics for Isabelle/Isar (ZF logic).

Copyright (C) 2005, 2006 Slawomir Kolodynski

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*)

header{*\isaheader{Group\_ZF\_2.thy}*}

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.*}

section{*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,\<one>)"

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;

section{*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``{\<one>}"

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;

section{*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,\<one>⟩ ∈ 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,\<one>⟩ ∈ r"

proof

assume "r``{a} = e"

with groupAssum assms have

"r``{\<one>} = 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 "⟨\<one>,a⟩ ∈ r" using eq_equiv_class

by simp;

with I show "⟨a,\<one>⟩ ∈ r" by (rule equiv_is_sym);

next assume "⟨a,\<one>⟩ ∈ r"

moreover from A1 A3 have "equiv(G,r)"

using IsAnormalSubgroup_def Group_ZF_2_4_L3

by simp;

ultimately have "r``{a} = r``{\<one>}"

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;

section{*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;