Theorem ecopoverg 6230

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 4481	. . . 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 6228	. . . 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 6229	. . . 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 495	. . . . . . . . . . 11 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> g e. S )
14		simplr 496	. . . . . . . . . . 11 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> h e. S )
15	12, 13, 14	caovcomd 5677	. . . . . . . . . 10 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> ( g .+ h ) = ( h .+ g ) )
16	1	ecopoveq 6224	. . . . . . . . . 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 2417	. . . . . . 7 |- A. g e. S A. h e. S <. g , h >. .~ <. g , h >.
20		breq12 3790	. . . . . . . . 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 4497	. . . . . . 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 2415	. . . . 5 |- ( f e. ( S X. S ) -> f .~ f )
25	24	a1i 9	. . . 4 |- ( T. -> ( f e. ( S X. S ) -> f .~ f ) )
26		opabssxp 4432	. . . . . . 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 3029	. . . . . 6 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
28	27	ssbri 3827	. . . . 5 |- ( f .~ f -> f ( ( S X. S ) X. ( S X. S ) ) f )
29		brxp 4393	. . . . . 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 6155	. 2 |- ( T. -> .~ Er ( S X. S ) )
34	33	trud 1293	1 |- .~ Er ( S X. S )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   T. wtru 1285   E.wex 1421   e. wcel 1433   A.wral 2348   <.cop 3401   class class class wbr 3785   {copab 3838   X. cxp 4361   Rel wrel 4368  (class class class)co 5532   Er wer 6126

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fv 4930  df-ov 5535  df-er 6129

This theorem is referenced by:  enqer  6548  enrer  6912