Theorem ecopovsymg 6228

Description: Assuming the operation F is commutative, show that the relation .~, specified by the first hypothesis, is symmetric. (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 ) )

Assertion

Ref	Expression
ecopovsymg	|- ( A .~ B -> B .~ A )

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

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

Proof of Theorem ecopovsymg

Dummy variables f g h t 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		opabssxp 4432	. . . . 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 ) ) ) } C_ ( ( S X. S ) X. ( S X. S ) )
3	1, 2	eqsstri 3029	. . . 4 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
4	3	brel 4410	. . 3 |- ( A .~ B -> ( A e. ( S X. S ) /\ B e. ( S X. S ) ) )
5		eqid 2081	. . . 4 |- ( S X. S ) = ( S X. S )
6		breq1 3788	. . . . 5 |- ( <. f , g >. = A -> ( <. f , g >. .~ <. h , t >. <-> A .~ <. h , t >. ) )
7		breq2 3789	. . . . 5 |- ( <. f , g >. = A -> ( <. h , t >. .~ <. f , g >. <-> <. h , t >. .~ A ) )
8	6, 7	bibi12d 233	. . . 4 |- ( <. f , g >. = A -> ( ( <. f , g >. .~ <. h , t >. <-> <. h , t >. .~ <. f , g >. ) <-> ( A .~ <. h , t >. <-> <. h , t >. .~ A ) ) )
9		breq2 3789	. . . . 5 |- ( <. h , t >. = B -> ( A .~ <. h , t >. <-> A .~ B ) )
10		breq1 3788	. . . . 5 |- ( <. h , t >. = B -> ( <. h , t >. .~ A <-> B .~ A ) )
11	9, 10	bibi12d 233	. . . 4 |- ( <. h , t >. = B -> ( ( A .~ <. h , t >. <-> <. h , t >. .~ A ) <-> ( A .~ B <-> B .~ A ) ) )
12		ecopoprg.com	. . . . . . . . 9 |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) = ( y .+ x ) )
13	12	adantl 271	. . . . . . . 8 |- ( ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )
14		simpll 495	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> f e. S )
15		simprr 498	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> t e. S )
16	13, 14, 15	caovcomd 5677	. . . . . . 7 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( f .+ t ) = ( t .+ f ) )
17		simplr 496	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> g e. S )
18		simprl 497	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> h e. S )
19	13, 17, 18	caovcomd 5677	. . . . . . 7 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( g .+ h ) = ( h .+ g ) )
20	16, 19	eqeq12d 2095	. . . . . 6 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( ( f .+ t ) = ( g .+ h ) <-> ( t .+ f ) = ( h .+ g ) ) )
21		eqcom 2083	. . . . . 6 |- ( ( t .+ f ) = ( h .+ g ) <-> ( h .+ g ) = ( t .+ f ) )
22	20, 21	syl6bb 194	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( ( f .+ t ) = ( g .+ h ) <-> ( h .+ g ) = ( t .+ f ) ) )
23	1	ecopoveq 6224	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( f .+ t ) = ( g .+ h ) ) )
24	1	ecopoveq 6224	. . . . . 6 |- ( ( ( h e. S /\ t e. S ) /\ ( f e. S /\ g e. S ) ) -> ( <. h , t >. .~ <. f , g >. <-> ( h .+ g ) = ( t .+ f ) ) )
25	24	ancoms 264	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. h , t >. .~ <. f , g >. <-> ( h .+ g ) = ( t .+ f ) ) )
26	22, 23, 25	3bitr4d 218	. . . 4 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> <. h , t >. .~ <. f , g >. ) )
27	5, 8, 11, 26	2optocl 4435	. . 3 |- ( ( A e. ( S X. S ) /\ B e. ( S X. S ) ) -> ( A .~ B <-> B .~ A ) )
28	4, 27	syl 14	. 2 |- ( A .~ B -> ( A .~ B <-> B .~ A ) )
29	28	ibi 174	1 |- ( A .~ B -> B .~ A )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   <.cop 3401   class class class wbr 3785   {copab 3838   X. cxp 4361  (class class class)co 5532

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-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  ecopoverg  6230