Theorem ecopovsym 6202

Description: Assuming the operation F is commutative, show that the relation .~, specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

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 ) ) ) }
ecopopr.com	|- ( x .+ y ) = ( y .+ x )

Assertion

Ref	Expression
ecopovsym	|- ( 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 ecopovsym

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 4414	. . . . 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 2975	. . . 4 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
4	3	brel 4392	. . 3 |- ( A .~ B -> ( A e. ( S X. S ) /\ B e. ( S X. S ) ) )
5		eqid 2040	. . . 4 |- ( S X. S ) = ( S X. S )
6		breq1 3767	. . . . 5 |- ( <. f , g >. = A -> ( <. f , g >. .~ <. h , t >. <-> A .~ <. h , t >. ) )
7		breq2 3768	. . . . 5 |- ( <. f , g >. = A -> ( <. h , t >. .~ <. f , g >. <-> <. h , t >. .~ A ) )
8	6, 7	bibi12d 224	. . . 4 |- ( <. f , g >. = A -> ( ( <. f , g >. .~ <. h , t >. <-> <. h , t >. .~ <. f , g >. ) <-> ( A .~ <. h , t >. <-> <. h , t >. .~ A ) ) )
9		breq2 3768	. . . . 5 |- ( <. h , t >. = B -> ( A .~ <. h , t >. <-> A .~ B ) )
10		breq1 3767	. . . . 5 |- ( <. h , t >. = B -> ( <. h , t >. .~ A <-> B .~ A ) )
11	9, 10	bibi12d 224	. . . 4 |- ( <. h , t >. = B -> ( ( A .~ <. h , t >. <-> <. h , t >. .~ A ) <-> ( A .~ B <-> B .~ A ) ) )
12	1	ecopoveq 6201	. . . . . 6 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( f .+ t ) = ( g .+ h ) ) )
13		vex 2560	. . . . . . . . 9 |- f e. _V
14		vex 2560	. . . . . . . . 9 |- t e. _V
15		ecopopr.com	. . . . . . . . 9 |- ( x .+ y ) = ( y .+ x )
16	13, 14, 15	caovcom 5658	. . . . . . . 8 |- ( f .+ t ) = ( t .+ f )
17		vex 2560	. . . . . . . . 9 |- g e. _V
18		vex 2560	. . . . . . . . 9 |- h e. _V
19	17, 18, 15	caovcom 5658	. . . . . . . 8 |- ( g .+ h ) = ( h .+ g )
20	16, 19	eqeq12i 2053	. . . . . . 7 |- ( ( f .+ t ) = ( g .+ h ) <-> ( t .+ f ) = ( h .+ g ) )
21		eqcom 2042	. . . . . . 7 |- ( ( t .+ f ) = ( h .+ g ) <-> ( h .+ g ) = ( t .+ f ) )
22	20, 21	bitri 173	. . . . . 6 |- ( ( f .+ t ) = ( g .+ h ) <-> ( h .+ g ) = ( t .+ f ) )
23	12, 22	syl6bb 185	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( h .+ g ) = ( t .+ f ) ) )
24	1	ecopoveq 6201	. . . . . 6 |- ( ( ( h e. S /\ t e. S ) /\ ( f e. S /\ g e. S ) ) -> ( <. h , t >. .~ <. f , g >. <-> ( h .+ g ) = ( t .+ f ) ) )
25	24	ancoms 255	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. h , t >. .~ <. f , g >. <-> ( h .+ g ) = ( t .+ f ) ) )
26	23, 25	bitr4d 180	. . . 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 4417	. . 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 165	1 |- ( A .~ B -> B .~ A )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   <.cop 3378   class class class wbr 3764   {copab 3817   X. cxp 4343  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  ecopover  6204