Theorem caovassg 5785

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 2456	. 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 5641	. . . . 5 |- ( x = A -> ( x F y ) = ( A F y ) )
4	3	oveq1d 5649	. . . 4 |- ( x = A -> ( ( x F y ) F z ) = ( ( A F y ) F z ) )
5		oveq1 5641	. . . 4 |- ( x = A -> ( x F ( y F z ) ) = ( A F ( y F z ) ) )
6	4, 5	eqeq12d 2102	. . 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 5642	. . . . 5 |- ( y = B -> ( A F y ) = ( A F B ) )
8	7	oveq1d 5649	. . . 4 |- ( y = B -> ( ( A F y ) F z ) = ( ( A F B ) F z ) )
9		oveq1 5641	. . . . 5 |- ( y = B -> ( y F z ) = ( B F z ) )
10	9	oveq2d 5650	. . . 4 |- ( y = B -> ( A F ( y F z ) ) = ( A F ( B F z ) ) )
11	8, 10	eqeq12d 2102	. . 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 5642	. . . 4 |- ( z = C -> ( ( A F B ) F z ) = ( ( A F B ) F C ) )
13		oveq2 5642	. . . . 5 |- ( z = C -> ( B F z ) = ( B F C ) )
14	13	oveq2d 5650	. . . 4 |- ( z = C -> ( A F ( B F z ) ) = ( A F ( B F C ) ) )
15	12, 14	eqeq12d 2102	. . 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 2736	. 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 275	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 102   /\ w3a 924   = wceq 1289   e. wcel 1438   A.wral 2359  (class class class)co 5634

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637

This theorem is referenced by:  caovassd  5786  caovass  5787  grprinvlem  5821  grprinvd  5822  grpridd  5823  seq3split  9872  iseqsplit  9873  iseqcaopr  9877