Theorem ecopoverg 6391

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 Jim Kingdon, 1-Sep-2019.)

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 ) ) ) }
ecopoprg.com	|- ( ( x e. S /\ y e. S ) -> ( x .+ y ) = ( y .+ x ) )
ecopoprg.cl	|- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )
ecopoprg.ass	|- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
ecopoprg.can	|- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )

Assertion

Ref	Expression
ecopoverg	|- .~ 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 ecopoverg

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 4563	. . . 4 |- Rel .~
3	2	a1i 9	. . 3 |- ( T. -> Rel .~ )
4		ecopoprg.com	. . . . 5 |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) = ( y .+ x ) )
5	1, 4	ecopovsymg 6389	. . . 4 |- ( f .~ g -> g .~ f )
6	5	adantl 271	. . 3 |- ( ( T. /\ f .~ g ) -> g .~ f )
7		ecopoprg.cl	. . . . 5 |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )
8		ecopoprg.ass	. . . . 5 |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
9		ecopoprg.can	. . . . 5 |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )
10	1, 4, 7, 8, 9	ecopovtrng 6390	. . . 4 |- ( ( f .~ g /\ g .~ h ) -> f .~ h )
11	10	adantl 271	. . 3 |- ( ( T. /\ ( f .~ g /\ g .~ h ) ) -> f .~ h )
12	4	adantl 271	. . . . . . . . . . 11 |- ( ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )
13		simpll 496	. . . . . . . . . . 11 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> g e. S )
14		simplr 497	. . . . . . . . . . 11 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> h e. S )
15	12, 13, 14	caovcomd 5801	. . . . . . . . . 10 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> ( g .+ h ) = ( h .+ g ) )
16	1	ecopoveq 6385	. . . . . . . . . 10 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> ( <. g , h >. .~ <. g , h >. <-> ( g .+ h ) = ( h .+ g ) ) )
17	15, 16	mpbird 165	. . . . . . . . 9 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> <. g , h >. .~ <. g , h >. )
18	17	anidms 389	. . . . . . . 8 |- ( ( g e. S /\ h e. S ) -> <. g , h >. .~ <. g , h >. )
19	18	rgen2a 2429	. . . . . . 7 |- A. g e. S A. h e. S <. g , h >. .~ <. g , h >.
20		breq12 3850	. . . . . . . . 9 |- ( ( f = <. g , h >. /\ f = <. g , h >. ) -> ( f .~ f <-> <. g , h >. .~ <. g , h >. ) )
21	20	anidms 389	. . . . . . . 8 |- ( f = <. g , h >. -> ( f .~ f <-> <. g , h >. .~ <. g , h >. ) )
22	21	ralxp 4579	. . . . . . 7 |- ( A. f e. ( S X. S ) f .~ f <-> A. g e. S A. h e. S <. g , h >. .~ <. g , h >. )
23	19, 22	mpbir 144	. . . . . 6 |- A. f e. ( S X. S ) f .~ f
24	23	rspec 2427	. . . . 5 |- ( f e. ( S X. S ) -> f .~ f )
25	24	a1i 9	. . . 4 |- ( T. -> ( f e. ( S X. S ) -> f .~ f ) )
26		opabssxp 4512	. . . . . . 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 ) )
27	1, 26	eqsstri 3056	. . . . . 6 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
28	27	ssbri 3887	. . . . 5 |- ( f .~ f -> f ( ( S X. S ) X. ( S X. S ) ) f )
29		brxp 4468	. . . . . 6 |- ( f ( ( S X. S ) X. ( S X. S ) ) f <-> ( f e. ( S X. S ) /\ f e. ( S X. S ) ) )
30	29	simplbi 268	. . . . 5 |- ( f ( ( S X. S ) X. ( S X. S ) ) f -> f e. ( S X. S ) )
31	28, 30	syl 14	. . . 4 |- ( f .~ f -> f e. ( S X. S ) )
32	25, 31	impbid1 140	. . 3 |- ( T. -> ( f e. ( S X. S ) <-> f .~ f ) )
33	3, 6, 11, 32	iserd 6316	. 2 |- ( T. -> .~ Er ( S X. S ) )
34	33	mptru 1298	1 |- .~ Er ( S X. S )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 924   = wceq 1289   T. wtru 1290   E.wex 1426   e. wcel 1438   A.wral 2359   <.cop 3449   class class class wbr 3845   {copab 3898   X. cxp 4436   Rel wrel 4443  (class class class)co 5652   Er wer 6287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fv 5023  df-ov 5655  df-er 6290

This theorem is referenced by:  enqer  6915  enrer  7279