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	qsid 4301	A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.)
|- (A/.`'E) = A
Theorem	ectocl 4302	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 4303	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 4304	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 4305	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 4306	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 4307	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 4308	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 4309	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 4310	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 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 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 4312	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 4313	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 4314	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption.
Theorem	th3qlem2 4315	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 4316	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 4317	Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs.
|- R e. V   &   |- Er R   &   |- dom R = (S X. S)   &   |- (