mmtheorems44 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4301-4400 - Page 44 of 107

Type	Label	Description
Statement
Theorem	ecopopreq 4301	This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation R (specified by the hypothesis) in terms of its operation F.
|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}   =>   |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> (AFD) = (BFC)))
Theorem	ecopoprdm 4302	Assuming the operation F is commutative, compute the domain the relation R specified by the first hypothesis.
|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}   &   |- (xFy) = (yFx)   =>   |- dom R = (S X. S)
Theorem	ecopoprsym 4303	Assuming the operation F is commutative, show that the relation R, specified by the first hypothesis, is symmetric.
|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}   &   |- (xFy) = (yFx)   &   |- B e. V   =>   |- (ARB -> BRA)
Theorem	ecopoprtrn 4304	Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation R, specified by the first hypothesis, is transitive.
|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}   &   |- (xFy) = (yFx)   &   |- ((x e. S /\ y e. S) -> (xFy) e. S)   &   |- ((xFy)Fz) = (xF(yFz))   &   |- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))   &   |- B e. V   &   |- C e. V   =>   |- ((ARB /\ BRC) -> ARC)
Theorem	ecopoprer 4305	Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation R, specified by the first hypothesis, is an equivalence relation.
|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}   &   |- (xFy) = (yFx)   &   |- ((x e. S /\ y e. S) -> (xFy) e. S)   &   |- ((xFy)Fz) = (xF(yFz))   &   |- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))   =>   |- Er R
Theorem	eceqopreq 4306	Equality of equivalence relation in terms of an operation.
|- B e. V   &   |- C e. V   &   |- D e. V   &   |- Er R   &   |- dom R = (S X. S)   &   |- dom F = (S X. S)   &   |- -. (/) e. S   &   |- ((x e. S /\ y e. S) -> (xFy) e. S)   &   |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> (AFD) = (BFC)))   =>   |- ((A e. S /\ C e. S) -> ([<.A, B>.]R = [<.C, D>.]R <-> (AFD) = (BFC)))
Theorem	th3qlem1 4307	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption.
Theorem	th3qlem2 4308	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption.
Theorem	th3qcor 4309	Corollary of Theorem 3Q of [Enderton] p. 60.
|- R e. V   &   |- Er R   &   |- dom R = (S X. S)   &   |- ((((w e. S /\ v e. S) /\ (u e. S /\ t e. S)) /\ ((s e. S /\ f e. S) /\ (g e. S /\ h e. S))) -> ((<.w, v>.R<.u, t>. /\ <.s, f>.R<.g, h>.) -> (<.w, v>.F<.s, f>.)R(<.u, t>.F<.g, h>.)))   &   |- G = {<.<.x, y>., z>. | ((x e. ((S X. S)/.R) /\ y e. ((S X. S)/.R)) /\ E.wE.vE.uE.t((x = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R))}   =>   |- Fun G
Theorem	th3q 4310	Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs.
|- R e. V   &   |- Er R   &   |- dom R = (S X. S)   &   |- ((((w e. S /\ v e. S) /\ (u e. S /\ t e. S)) /\ ((s e. S /\ f e. S) /\ (g e. S /\ h e. S))) -> ((<.w, v>.R<.u, t>. /\ <.s, f>.R<.g, h>.) -> (<.w, v>.F<.s, f>.)R(<.u, t>.F<.g, h>.)))   &   |- G = {<.<.x, y>., z>. | ((x e. ((S X. S)/.R) /\ y e. ((S X. S)/.R)) /\ E.wE.vE.uE.t((x = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R))}   =>   |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> ([<.A, B>.]RG[<.C, D>.]R) = [(<.A, B>.F<.C, D>.)]R)
Theorem	oprec 4311	Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. (See set.mm for additional comments for the hypotheses.)
|- H e. V   &   |- K e. V   &   |- L e. V   &   |- R e. V   &   |- Er R   &   |- dom R = (S X. S)   &   |- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ ph))}   &   |- (((z = a /\ w = b) /\ (v = c /\ u = d)) -> (ph <-> ps))   &   |- (((z = g /\ w = h) /\ (v = t /\ u = s)) -> (ph <-> ch))   &   |- G = {<.<.x, y>., z>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = J))}   &   |- (((w = a /\ v = b) /\ (u = g /\ f = h)) -> J = K)   &   |- (((w = c /\ v = d) /\ (u = t /\ f = s)) -> J = L)   &   |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> J = H)   &   |- F = {<.<.x, y>., z>. | ((x e. Q /\ y e. Q) /\ E.aE.bE.cE.d((x = [<.a, b>.]R /\ y = [<.c, d>.]R) /\ z = [(<.a, b>.G<.c, d>.)]R))}   &   |- Q = ((S X. S)/.R)   &   |- ((((a e. S /\ b e. S) /\ (c e. S /\ d e. S)) /\ ((g e. S /\ h e. S) /\ (t e. S /\ s e. S))) -> ((ps /\ ch) -> KRL))   =>   |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> ([<.A, B>.]RF[<.C, D>.]R) = [H]R)
Theorem	ecoprcom 4312	Lemma used to transfer a commutative law via an equivalence relation.
|-