Theory Group_ZF_2

theory Group_ZF_2
imports AbelianGroup_ZF EquivClass1
(* 
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;