Theorem ecopovsymg 6870

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 4826	. . . . 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 3272	. . . 4 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
4	3	brel 4804	. . 3 |- ( A .~ B -> ( A e. ( S X. S ) /\ B e. ( S X. S ) ) )
5		eqid 2234	. . . 4 |- ( S X. S ) = ( S X. S )
6		breq1 4114	. . . . 5 |- ( <. f , g >. = A -> ( <. f , g >. .~ <. h , t >. <-> A .~ <. h , t >. ) )
7		breq2 4115	. . . . 5 |- ( <. f , g >. = A -> ( <. h , t >. .~ <. f , g >. <-> <. h , t >. .~ A ) )
8	6, 7	bibi12d 235	. . . 4 |- ( <. f , g >. = A -> ( ( <. f , g >. .~ <. h , t >. <-> <. h , t >. .~ <. f , g >. ) <-> ( A .~ <. h , t >. <-> <. h , t >. .~ A ) ) )
9		breq2 4115	. . . . 5 |- ( <. h , t >. = B -> ( A .~ <. h , t >. <-> A .~ B ) )
10		breq1 4114	. . . . 5 |- ( <. h , t >. = B -> ( <. h , t >. .~ A <-> B .~ A ) )
11	9, 10	bibi12d 235	. . . 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 277	. . . . . . . 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 527	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> f e. S )
15		simprr 533	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> t e. S )
16	13, 14, 15	caovcomd 6213	. . . . . . 7 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( f .+ t ) = ( t .+ f ) )
17		simplr 529	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> g e. S )
18		simprl 531	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> h e. S )
19	13, 17, 18	caovcomd 6213	. . . . . . 7 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( g .+ h ) = ( h .+ g ) )
20	16, 19	eqeq12d 2249	. . . . . 6 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( ( f .+ t ) = ( g .+ h ) <-> ( t .+ f ) = ( h .+ g ) ) )
21		eqcom 2236	. . . . . 6 |- ( ( t .+ f ) = ( h .+ g ) <-> ( h .+ g ) = ( t .+ f ) )
22	20, 21	bitrdi 196	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( ( f .+ t ) = ( g .+ h ) <-> ( h .+ g ) = ( t .+ f ) ) )
23	1	ecopoveq 6866	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( f .+ t ) = ( g .+ h ) ) )
24	1	ecopoveq 6866	. . . . . 6 |- ( ( ( h e. S /\ t e. S ) /\ ( f e. S /\ g e. S ) ) -> ( <. h , t >. .~ <. f , g >. <-> ( h .+ g ) = ( t .+ f ) ) )
25	24	ancoms 268	. . . . 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 220	. . . 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 4829	. . 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 176	1 |- ( A .~ B -> B .~ A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   <.cop 3694   class class class wbr 4111   {copab 4172   X. cxp 4749  (class class class)co 6052

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-xp 4757  df-iota 5314  df-fv 5362  df-ov 6055

This theorem is referenced by:  ecopoverg  6872