Theorem ecopovsymg 6612

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 4685	. . . . 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 3179	. . . 4 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
4	3	brel 4663	. . 3 |- ( A .~ B -> ( A e. ( S X. S ) /\ B e. ( S X. S ) ) )
5		eqid 2170	. . . 4 |- ( S X. S ) = ( S X. S )
6		breq1 3992	. . . . 5 |- ( <. f , g >. = A -> ( <. f , g >. .~ <. h , t >. <-> A .~ <. h , t >. ) )
7		breq2 3993	. . . . 5 |- ( <. f , g >. = A -> ( <. h , t >. .~ <. f , g >. <-> <. h , t >. .~ A ) )
8	6, 7	bibi12d 234	. . . 4 |- ( <. f , g >. = A -> ( ( <. f , g >. .~ <. h , t >. <-> <. h , t >. .~ <. f , g >. ) <-> ( A .~ <. h , t >. <-> <. h , t >. .~ A ) ) )
9		breq2 3993	. . . . 5 |- ( <. h , t >. = B -> ( A .~ <. h , t >. <-> A .~ B ) )
10		breq1 3992	. . . . 5 |- ( <. h , t >. = B -> ( <. h , t >. .~ A <-> B .~ A ) )
11	9, 10	bibi12d 234	. . . 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 275	. . . . . . . 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 524	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> f e. S )
15		simprr 527	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> t e. S )
16	13, 14, 15	caovcomd 6009	. . . . . . 7 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( f .+ t ) = ( t .+ f ) )
17		simplr 525	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> g e. S )
18		simprl 526	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> h e. S )
19	13, 17, 18	caovcomd 6009	. . . . . . 7 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( g .+ h ) = ( h .+ g ) )
20	16, 19	eqeq12d 2185	. . . . . 6 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( ( f .+ t ) = ( g .+ h ) <-> ( t .+ f ) = ( h .+ g ) ) )
21		eqcom 2172	. . . . . 6 |- ( ( t .+ f ) = ( h .+ g ) <-> ( h .+ g ) = ( t .+ f ) )
22	20, 21	bitrdi 195	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( ( f .+ t ) = ( g .+ h ) <-> ( h .+ g ) = ( t .+ f ) ) )
23	1	ecopoveq 6608	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( f .+ t ) = ( g .+ h ) ) )
24	1	ecopoveq 6608	. . . . . 6 |- ( ( ( h e. S /\ t e. S ) /\ ( f e. S /\ g e. S ) ) -> ( <. h , t >. .~ <. f , g >. <-> ( h .+ g ) = ( t .+ f ) ) )
25	24	ancoms 266	. . . . 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 219	. . . 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 4688	. . 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 175	1 |- ( A .~ B -> B .~ A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   <.cop 3586   class class class wbr 3989   {copab 4049   X. cxp 4609  (class class class)co 5853

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-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  ecopoverg  6614