Theorem ecopover 6611

Description: 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 .~, specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ecopopr.1	|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }
ecopopr.com	|- ( x .+ y ) = ( y .+ x )
ecopopr.cl	|- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )
ecopopr.ass	|- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) )
ecopopr.can	|- ( ( x e. S /\ y e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )

Assertion

Ref	Expression
ecopover	|- .~ Er ( S X. S )

Distinct variable groups:   x, y, z, w, v, u, .+   x, S, y, z, w, v, u

Allowed substitution hints:   .~ ( x, y, z, w, v, u)

Proof of Theorem ecopover

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecopopr.1	. . . . 5 |- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }
2	1	relopabi 4737	. . . 4 |- Rel .~
3	2	a1i 9	. . 3 |- ( T. -> Rel .~ )
4		ecopopr.com	. . . . 5 |- ( x .+ y ) = ( y .+ x )
5	1, 4	ecopovsym 6609	. . . 4 |- ( f .~ g -> g .~ f )
6	5	adantl 275	. . 3 |- ( ( T. /\ f .~ g ) -> g .~ f )
7		ecopopr.cl	. . . . 5 |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )
8		ecopopr.ass	. . . . 5 |- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) )
9		ecopopr.can	. . . . 5 |- ( ( x e. S /\ y e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )
10	1, 4, 7, 8, 9	ecopovtrn 6610	. . . 4 |- ( ( f .~ g /\ g .~ h ) -> f .~ h )
11	10	adantl 275	. . 3 |- ( ( T. /\ ( f .~ g /\ g .~ h ) ) -> f .~ h )
12		vex 2733	. . . . . . . . . . 11 |- g e. _V
13		vex 2733	. . . . . . . . . . 11 |- h e. _V
14	12, 13, 4	caovcom 6010	. . . . . . . . . 10 |- ( g .+ h ) = ( h .+ g )
15	1	ecopoveq 6608	. . . . . . . . . 10 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> ( <. g , h >. .~ <. g , h >. <-> ( g .+ h ) = ( h .+ g ) ) )
16	14, 15	mpbiri 167	. . . . . . . . 9 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> <. g , h >. .~ <. g , h >. )
17	16	anidms 395	. . . . . . . 8 |- ( ( g e. S /\ h e. S ) -> <. g , h >. .~ <. g , h >. )
18	17	rgen2a 2524	. . . . . . 7 |- A. g e. S A. h e. S <. g , h >. .~ <. g , h >.
19		breq12 3994	. . . . . . . . 9 |- ( ( f = <. g , h >. /\ f = <. g , h >. ) -> ( f .~ f <-> <. g , h >. .~ <. g , h >. ) )
20	19	anidms 395	. . . . . . . 8 |- ( f = <. g , h >. -> ( f .~ f <-> <. g , h >. .~ <. g , h >. ) )
21	20	ralxp 4754	. . . . . . 7 |- ( A. f e. ( S X. S ) f .~ f <-> A. g e. S A. h e. S <. g , h >. .~ <. g , h >. )
22	18, 21	mpbir 145	. . . . . 6 |- A. f e. ( S X. S ) f .~ f
23	22	rspec 2522	. . . . 5 |- ( f e. ( S X. S ) -> f .~ f )
24	23	a1i 9	. . . 4 |- ( T. -> ( f e. ( S X. S ) -> f .~ f ) )
25		opabssxp 4685	. . . . . . 7 |- { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) } C_ ( ( S X. S ) X. ( S X. S ) )
26	1, 25	eqsstri 3179	. . . . . 6 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
27	26	ssbri 4033	. . . . 5 |- ( f .~ f -> f ( ( S X. S ) X. ( S X. S ) ) f )
28		brxp 4642	. . . . . 6 |- ( f ( ( S X. S ) X. ( S X. S ) ) f <-> ( f e. ( S X. S ) /\ f e. ( S X. S ) ) )
29	28	simplbi 272	. . . . 5 |- ( f ( ( S X. S ) X. ( S X. S ) ) f -> f e. ( S X. S ) )
30	27, 29	syl 14	. . . 4 |- ( f .~ f -> f e. ( S X. S ) )
31	24, 30	impbid1 141	. . 3 |- ( T. -> ( f e. ( S X. S ) <-> f .~ f ) )
32	3, 6, 11, 31	iserd 6539	. 2 |- ( T. -> .~ Er ( S X. S ) )
33	32	mptru 1357	1 |- .~ Er ( S X. S )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   T. wtru 1349   E.wex 1485   e. wcel 2141   A.wral 2448   <.cop 3586   class class class wbr 3989   {copab 4049   X. cxp 4609   Rel wrel 4616  (class class class)co 5853   Er wer 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fv 5206  df-ov 5856  df-er 6513

This theorem is referenced by: (None)