Theorem ecopovsymg 6521

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 4608	. . . . 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 3124	. . . 4 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
4	3	brel 4586	. . 3 |- ( A .~ B -> ( A e. ( S X. S ) /\ B e. ( S X. S ) ) )
5		eqid 2137	. . . 4 |- ( S X. S ) = ( S X. S )
6		breq1 3927	. . . . 5 |- ( <. f , g >. = A -> ( <. f , g >. .~ <. h , t >. <-> A .~ <. h , t >. ) )
7		breq2 3928	. . . . 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 3928	. . . . 5 |- ( <. h , t >. = B -> ( A .~ <. h , t >. <-> A .~ B ) )
10		breq1 3927	. . . . 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 518	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> f e. S )
15		simprr 521	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> t e. S )
16	13, 14, 15	caovcomd 5920	. . . . . . 7 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( f .+ t ) = ( t .+ f ) )
17		simplr 519	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> g e. S )
18		simprl 520	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> h e. S )
19	13, 17, 18	caovcomd 5920	. . . . . . 7 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( g .+ h ) = ( h .+ g ) )
20	16, 19	eqeq12d 2152	. . . . . 6 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( ( f .+ t ) = ( g .+ h ) <-> ( t .+ f ) = ( h .+ g ) ) )
21		eqcom 2139	. . . . . 6 |- ( ( t .+ f ) = ( h .+ g ) <-> ( h .+ g ) = ( t .+ f ) )
22	20, 21	syl6bb 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 6517	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( f .+ t ) = ( g .+ h ) ) )
24	1	ecopoveq 6517	. . . . . 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 4611	. . 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 1331   E.wex 1468   e. wcel 1480   <.cop 3525   class class class wbr 3924   {copab 3983   X. cxp 4532  (class class class)co 5767

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-iota 5083  df-fv 5126  df-ov 5770

This theorem is referenced by:  ecopoverg  6523