Theorem caovassg 5659

Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)

Hypothesis

Ref	Expression
caovassg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )

Assertion

Ref	Expression
caovassg	|- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   ph, x, y, z   x, F, y, z   x, S, y, z

Proof of Theorem caovassg

Step	Hyp	Ref	Expression
1		caovassg.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
2	1	ralrimivvva 2402	. 2 |- ( ph -> A. x e. S A. y e. S A. z e. S ( ( x F y ) F z ) = ( x F ( y F z ) ) )
3		oveq1 5519	. . . . 5 |- ( x = A -> ( x F y ) = ( A F y ) )
4	3	oveq1d 5527	. . . 4 |- ( x = A -> ( ( x F y ) F z ) = ( ( A F y ) F z ) )
5		oveq1 5519	. . . 4 |- ( x = A -> ( x F ( y F z ) ) = ( A F ( y F z ) ) )
6	4, 5	eqeq12d 2054	. . 3 |- ( x = A -> ( ( ( x F y ) F z ) = ( x F ( y F z ) ) <-> ( ( A F y ) F z ) = ( A F ( y F z ) ) ) )
7		oveq2 5520	. . . . 5 |- ( y = B -> ( A F y ) = ( A F B ) )
8	7	oveq1d 5527	. . . 4 |- ( y = B -> ( ( A F y ) F z ) = ( ( A F B ) F z ) )
9		oveq1 5519	. . . . 5 |- ( y = B -> ( y F z ) = ( B F z ) )
10	9	oveq2d 5528	. . . 4 |- ( y = B -> ( A F ( y F z ) ) = ( A F ( B F z ) ) )
11	8, 10	eqeq12d 2054	. . 3 |- ( y = B -> ( ( ( A F y ) F z ) = ( A F ( y F z ) ) <-> ( ( A F B ) F z ) = ( A F ( B F z ) ) ) )
12		oveq2 5520	. . . 4 |- ( z = C -> ( ( A F B ) F z ) = ( ( A F B ) F C ) )
13		oveq2 5520	. . . . 5 |- ( z = C -> ( B F z ) = ( B F C ) )
14	13	oveq2d 5528	. . . 4 |- ( z = C -> ( A F ( B F z ) ) = ( A F ( B F C ) ) )
15	12, 14	eqeq12d 2054	. . 3 |- ( z = C -> ( ( ( A F B ) F z ) = ( A F ( B F z ) ) <-> ( ( A F B ) F C ) = ( A F ( B F C ) ) ) )
16	6, 11, 15	rspc3v 2665	. 2 |- ( ( A e. S /\ B e. S /\ C e. S ) -> ( A. x e. S A. y e. S A. z e. S ( ( x F y ) F z ) = ( x F ( y F z ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) ) )
17	2, 16	mpan9 265	1 |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393   A.wral 2306  (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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  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-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  caovassd  5660  caovass  5661  grprinvlem  5695  grprinvd  5696  grpridd  5697  iseqsplit  9238  iseqcaopr  9242