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	ectocl 4301	Implicit substitution of class for equivalence class.
|- S = (B/.R)   &   |- ([x]R = A -> (ph <-> ps))   &   |- (x e. B -> ph)   =>   |- (A e. S -> ps)
Theorem	ecoptocl 4302	Implicit substitution of class for equivalence class of ordered pair.
|- S = ((B X. C)/.R)   &   |- ([<.x, y>.]R = A -> (ph <-> ps))   &   |- ((x e. B /\ y e. C) -> ph)   =>   |- (A e. S -> ps)
Theorem	2ecoptocl 4303	Implicit substitution of classes for equivalence classes of ordered pairs.
|- S = ((C X. D)/.R)   &   |- ([<.x, y>.]R = A -> (ph <-> ps))   &   |- ([<.z, w>.]R = B -> (ps <-> ch))   &   |- (((x e. C /\ y e. D) /\ (z e. C /\ w e. D)) -> ph)   =>   |- ((A e. S /\ B e. S) -> ch)
Theorem	3ecoptocl 4304	Implicit substitution of classes for equivalence classes of ordered pairs.
|- S = ((D X. D)/.R)   &   |- ([<.x, y>.]R = A -> (ph <-> ps))   &   |- ([<.z, w>.]R = B -> (ps <-> ch))   &   |- ([<.v, u>.]R = C -> (ch <-> th))   &   |- (((x e. D /\ y e. D) /\ (z e. D /\ w e. D) /\ (v e. D /\ u e. D)) -> ph)   =>   |- ((A e. S /\ B e. S /\ C e. S) -> th)
Theorem	brecop 4305	Binary relation on a quotient set. Lemma for real number construction.
|- S e. V   &   |- Er S   &   |- dom S = (G X. G)   &   |- H = ((G X. G)/.S)   &   |- R = {<.x, y>. | ((x e. H /\ y e. H) /\ E.zE.wE.vE.u((x = [<.z, w>.]S /\ y = [<.v, u>.]S) /\ ph))}   &   |- ((((z e. G /\ w e. G) /\ (A e. G /\ B e. G)) /\ ((v e. G /\ u e. G) /\ (C e. G /\ D e. G))) -> (([<.z, w>.]S = [<.A, B>.]S /\ [<.v, u>.]S = [<.C, D>.]S) -> (ph <-> ps)))   =>   |- (((A e. G /\ B e. G) /\ (C e. G /\ D e. G)) -> ([<.A, B>.]SR[<.C, D>.]S <-> ps))
Theorem	brecop2 4306	Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis.
|- S e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   &   |- dom S = (G X. G)   &   |- H = ((G X. G)/.S)   &   |- R (_ (H X. H)   &   |- Q (_ (G X. G)   &   |- -. (/) e. G   &   |- dom F = (G X. G)   &   |- (((A e. G /\ B e. G) /\ (C e. G /\ D e. G)) -> ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC)))   =>   |- ([<.A, B>.]SR[<.C, D>.]S <-> (AFD)Q(BFC))
Theorem	ecopopreq 4307	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 4308	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 4309	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 4310	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 4311	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 4312	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 4313	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption.
Theorem	th3qlem2 4314	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 4315	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 4316	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