Theorem ecopoverg 6800

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 4853	. . . 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 6798	. . . 4 |- ( f .~ g -> g .~ f )
6	5	adantl 277	. . 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 6799	. . . 4 |- ( ( f .~ g /\ g .~ h ) -> f .~ h )
11	10	adantl 277	. . 3 |- ( ( T. /\ ( f .~ g /\ g .~ h ) ) -> f .~ h )
12	4	adantl 277	. . . . . . . . . . 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 527	. . . . . . . . . . 11 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> g e. S )
14		simplr 528	. . . . . . . . . . 11 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> h e. S )
15	12, 13, 14	caovcomd 6174	. . . . . . . . . 10 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> ( g .+ h ) = ( h .+ g ) )
16	1	ecopoveq 6794	. . . . . . . . . 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 167	. . . . . . . . 9 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> <. g , h >. .~ <. g , h >. )
18	17	anidms 397	. . . . . . . 8 |- ( ( g e. S /\ h e. S ) -> <. g , h >. .~ <. g , h >. )
19	18	rgen2a 2584	. . . . . . 7 |- A. g e. S A. h e. S <. g , h >. .~ <. g , h >.
20		breq12 4091	. . . . . . . . 9 |- ( ( f = <. g , h >. /\ f = <. g , h >. ) -> ( f .~ f <-> <. g , h >. .~ <. g , h >. ) )
21	20	anidms 397	. . . . . . . 8 |- ( f = <. g , h >. -> ( f .~ f <-> <. g , h >. .~ <. g , h >. ) )
22	21	ralxp 4871	. . . . . . 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 146	. . . . . 6 |- A. f e. ( S X. S ) f .~ f
24	23	rspec 2582	. . . . 5 |- ( f e. ( S X. S ) -> f .~ f )
25	24	a1i 9	. . . 4 |- ( T. -> ( f e. ( S X. S ) -> f .~ f ) )
26		opabssxp 4798	. . . . . . 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 3257	. . . . . 6 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
28	27	ssbri 4131	. . . . 5 |- ( f .~ f -> f ( ( S X. S ) X. ( S X. S ) ) f )
29		brxp 4754	. . . . . 6 |- ( f ( ( S X. S ) X. ( S X. S ) ) f <-> ( f e. ( S X. S ) /\ f e. ( S X. S ) ) )
30	29	simplbi 274	. . . . 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 142	. . 3 |- ( T. -> ( f e. ( S X. S ) <-> f .~ f ) )
33	3, 6, 11, 32	iserd 6723	. 2 |- ( T. -> .~ Er ( S X. S ) )
34	33	mptru 1404	1 |- .~ Er ( S X. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   T. wtru 1396   E.wex 1538   e. wcel 2200   A.wral 2508   <.cop 3670   class class class wbr 4086   {copab 4147   X. cxp 4721   Rel wrel 4728  (class class class)co 6013   Er wer 6694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fv 5332  df-ov 6016  df-er 6697

This theorem is referenced by:  enqer  7568  enrer  7945