mmtheorems43 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4201-4300 - Page 43 of 105

Type	Label	Description
Statement
Definition	df-ec 4201	Define the R -coset of A. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of A modulo R when R is an equivalence relation. The Definition of [Enderton] p. 57 uses the notation [A] (subscript) R, although we simply follow the brackets by R since we don't have subscripts. For an alternate definition, see ec2 4202.
|- [A]R = (R"{A})
Theorem	ec2 4202	Alternate definition of R-coset of A. Definition 34 of [Suppes] p. 81.
|- A e. V   =>   |- [A]R = {y | ARy}
Theorem	ecexg 4203	An equivalence class modulo a set is a set.
|- (R e. B -> [A]R e. V)
Definition	df-qs 4204	Define quotient set. R is usually an equivalence relation. Definition of [Enderton] p. 58.
|- (A/.R) = {y | E.x e. A y = [x]R}
Theorem	ereq 4205	Equality theorem for equivalence predicate.
|- (R = S -> (Er R <-> Er S))
Theorem	ster 4206	A symmetric, transitive relation is an equivalence relation.
|- (xRy -> yRx)   &   |- ((xRy /\ yRz) -> xRz)   =>   |- Er R
Theorem	ider 4207	The identity relation is an equivalence relation.
|- Er I
Theorem	eqerlem 4208	Lemma for eqer 4209.
Theorem	eqer 4209	Equivalence relation involving equality of dependent classes A(x) and B(y).
|- (x = y -> A = B)   &   |- R = {<.x, y>. | A = B}   =>   |- Er R
Theorem	ersym 4210	An equivalence relation is symmetric.
|- A e. V   &   |- B e. V   &   |- Er R   =>   |- (ARB -> BRA)
Theorem	ersymb 4211	An equivalence relation is symmetric.
|- A e. V   &   |- B e. V   &   |- Er R   =>   |- (ARB <-> BRA)
Theorem	ertr 4212	An equivalence relation is transitive.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- Er R   =>   |- ((ARB /\ BRC) -> ARC)
Theorem	erref 4213	An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56.
|- Er R   =>   |- (A e. (dom R u. ran R) -> ARA)
Theorem	erdmrn 4214	The range and domain of an equivalence relation are equal.
|- Er R   =>   |- dom R = ran R
Theorem	eceq1 4215	Equality theorem for equivalence class.
|- (A = B -> [C]A = [C]B)
Theorem	eceq2 4216	Equality theorem for equivalence class.
|- (A = B -> [A]C = [B]C)
Theorem	elec 4217	Membership in an equivalence class. Theorem 72 of [Suppes] p. 82.
|- A e. V   &   |- B e. V   =>   |- (A e. [B]R <-> BRA)
Theorem	ecdmn0 4218	An equivalence class is not empty in its domain.
|- A e. V   =>   |- (A e. dom R <-> -. [A]R = (/))
Theorem	erthi 4219	Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57.
|- A e. V   &   |- B e. V   &   |- Er R   =>   |- (ARB -> [A]R = [B]R)
Theorem	erth 4220	Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82.
|- B e. V   &   |- Er R   =>   |- (A e. (dom R u. ran R) -> ([A]R = [B]R <-> ARB))
Theorem	erthdm 4221	Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership in the domain instead of just the field.
|- B e. V   &   |- Er R   =>   |- (A e. dom R -> ([A]R = [B]R <-> ARB))
Theorem	erthdmr 4222	Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain.
|- A e. V   &   |- B e. V   &   |- Er R   =>   |- (B e. dom R -> ([A]R = [B]R <-> ARB))
Theorem	ereldm 4223	Equality of equivalence classes implies equivalence of domain membership.
|- A e. V   &   |- B e. V   &   |- Er R   &   |- dom R = D   =>   |- ([A]R = [B]R -> (A e. D <-> B e. D))
Theorem	erdisj 4224	Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83.
|- A e. V   &   |- B e. V   &   |- Er R   =>   |- ([A]R = [B]R \/ ([A]R i^i [B]R) = (/))
Theorem	ecidsn 4225	An equivalence class modulo the identity relation is a singleton.
|- [A]I = {A}
Theorem	qseq1 4226	Equality theorem for quotient set.
|- (A = B -> (A/.C) = (B/.C))
Theorem	qseq2 4227	Equality theorem for quotient set.
|- (A = B -> (C/.A) = (C/.B))
Theorem	elqs 4228	Membership in a quotient set.
|- B e. V   =>   |- (B e. (A/.R) <-> E.x(x e. A /\ B = [x]R))
Theorem	elqsi 4229	Membership in a quotient set.
|- (B e. (A/.R) -> E.x(x e. A /\ B = [x]R))
Theorem	ecelqsi 4230	Membership of an equivalence class in a quotient set.
|- R e. V   =>   |- (B e. A -> [B]R e. (A/.R))
Theorem	ecopqsi 4231	"Closure" law for equivalence class of ordered pairs.
|- R e. V   &   |- S = ((A X. A)/.R)   =>   |- ((B e. A /\ C e. A) -> [<.B, C>.]R e. S)
Theorem	qsexg 4232	A quotient set exists. (Contributed by FL, 19-May-2007.)
|- (A e. V -> (A/.R) e. V)
Theorem	qsex 4233	A quotient set exists.
|- A e. V   =>   |- (A/.R) e. V
Theorem	snec 4234	The singleton of an equivalence class.
|- A e. V   =>   |- {[A]R} = ({A}/.R)
Theorem	ecqs 4235	Equivalence class in terms of quotient set.
|- A e. V   &   |- R e. V   =>   |- [A]R = U.({A}/.R)
Theorem	0nelqs 4236	A quotient set doesn't contain the empty set.
|- dom R = A   =>   |- -. (/) e. (A/.R)
Theorem	ecelqsdm 4237	Membership of an equivalence class in a quotient set.
|- B e. V   &   |- dom R = A   =>   |- ([B]R e. (A/.R) -> B e. A)
Theorem	ecid 4238	A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.)
|- A e. V   =>   |- [A]`'E = A
Theorem	qsid 4239	A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.)
|- (A/.`'E) = A
Theorem	ectocl 4240	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 4241	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 4242	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 4243	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 4244	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 4245	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 4246	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.