(*

This file is a part of IsarMathLib -

a library of formalized mathematics written for Isabelle/Isar.

Copyright (C) 2008 Seo Sanghyeon

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*)

header{*\isaheader{DirectProduct\_ZF.thy}*}

theory DirectProduct_ZF imports func_ZF

begin

text{*This theory considers the direct product of binary operations.

Contributed by Seo Sanghyeon. *}

section{*Definition*}

text{*In group theory the notion of direct product provides a natural

way of creating a new group from two given groups.*}

text{*Given $(G,\cdot)$ and $(H,\circ)$

a new operation $(G\times H, \times )$ is defined as

$(g, h) \times (g', h') = (g \cdot g', h \circ h')$. *}

definition

"DirectProduct(P,Q,G,H) ≡

{⟨x,⟨P`⟨fst(fst(x)),fst(snd(x))⟩ , Q`⟨snd(fst(x)),snd(snd(x))⟩⟩⟩.

x ∈ (G×H)×(G×H)}";

text{*We define a context called @{text "direct0"} which

holds an assumption that $P, Q$ are binary operations on

$G,H$, resp. and denotes $R$ as the direct product of

$(G,P)$ and $(H,Q)$.*}

locale direct0 =

fixes P Q G H

assumes Pfun: "P : G×G->G"

assumes Qfun: "Q : H×H->H"

fixes R

defines Rdef [simp]: "R ≡ DirectProduct(P,Q,G,H)";

text{*The direct product of binary operations is a binary operation.*}

lemma (in direct0) DirectProduct_ZF_1_L1:

shows "R : (G×H)×(G×H)->G×H"

proof -

from Pfun Qfun have "∀x∈(G×H)×(G×H).

⟨P`⟨fst(fst(x)),fst(snd(x))⟩,Q`⟨snd(fst(x)),snd(snd(x))⟩⟩ ∈ G×H"

by auto;

then show ?thesis using ZF_fun_from_total DirectProduct_def

by simp;

qed;

text{*And it has the intended value.*}

lemma (in direct0) DirectProduct_ZF_1_L2:

shows "∀x∈(G×H). ∀y∈(G×H).

R`⟨x,y⟩ = ⟨P`⟨fst(x),fst(y)⟩,Q`⟨snd(x),snd(y)⟩⟩"

using DirectProduct_def DirectProduct_ZF_1_L1 ZF_fun_from_tot_val

by simp;

text{*And the value belongs to the set the operation is defined on.*}

lemma (in direct0) DirectProduct_ZF_1_L3:

shows "∀x∈(G×H). ∀y∈(G×H). R`⟨x,y⟩ ∈ G×H"

using DirectProduct_ZF_1_L1 by simp;

section{*Associative and commutative operations*}

text{*If P and Q are both associative or commutative operations,

the direct product of P and Q has the same property.*}

text{*Direct product of commutative operations is commutative.*}

lemma (in direct0) DirectProduct_ZF_2_L1:

assumes "P {is commutative on} G" and "Q {is commutative on} H"

shows "R {is commutative on} G×H"

proof -

from assms have "∀x∈(G×H). ∀y∈(G×H). R`⟨x,y⟩ = R`⟨y,x⟩"

using DirectProduct_ZF_1_L2 IsCommutative_def by simp

then show ?thesis using IsCommutative_def by simp

qed;

text{*Direct product of associative operations is associative.*}

lemma (in direct0) DirectProduct_ZF_2_L2:

assumes "P {is associative on} G" and "Q {is associative on} H"

shows "R {is associative on} G×H"

proof -

have "∀x∈G×H. ∀y∈G×H. ∀z∈G×H. R`⟨R`⟨x,y⟩,z⟩ =

⟨P`⟨P`⟨fst(x),fst(y)⟩,fst(z)⟩,Q`⟨Q`⟨snd(x),snd(y)⟩,snd(z)⟩⟩"

using DirectProduct_ZF_1_L2 DirectProduct_ZF_1_L3

by auto;

moreover have "∀x∈G×H. ∀y∈G×H. ∀z∈G×H. R`⟨x,R`⟨y,z⟩⟩ =

⟨P`⟨fst(x),P`⟨fst(y),fst(z)⟩⟩,Q`⟨snd(x),Q`⟨snd(y),snd(z)⟩⟩⟩"

using DirectProduct_ZF_1_L2 DirectProduct_ZF_1_L3 by auto;

ultimately have "∀x∈G×H. ∀y∈G×H. ∀z∈G×H. R`⟨R`⟨x,y⟩,z⟩ = R`⟨x,R`⟨y,z⟩⟩"

using assms IsAssociative_def by simp

then show ?thesis

using DirectProduct_ZF_1_L1 IsAssociative_def by simp

qed;

end