Theorem caovassg 6128

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 2591	. 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 5974	. . . . 5 |- ( x = A -> ( x F y ) = ( A F y ) )
4	3	oveq1d 5982	. . . 4 |- ( x = A -> ( ( x F y ) F z ) = ( ( A F y ) F z ) )
5		oveq1 5974	. . . 4 |- ( x = A -> ( x F ( y F z ) ) = ( A F ( y F z ) ) )
6	4, 5	eqeq12d 2222	. . 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 5975	. . . . 5 |- ( y = B -> ( A F y ) = ( A F B ) )
8	7	oveq1d 5982	. . . 4 |- ( y = B -> ( ( A F y ) F z ) = ( ( A F B ) F z ) )
9		oveq1 5974	. . . . 5 |- ( y = B -> ( y F z ) = ( B F z ) )
10	9	oveq2d 5983	. . . 4 |- ( y = B -> ( A F ( y F z ) ) = ( A F ( B F z ) ) )
11	8, 10	eqeq12d 2222	. . 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 5975	. . . 4 |- ( z = C -> ( ( A F B ) F z ) = ( ( A F B ) F C ) )
13		oveq2 5975	. . . . 5 |- ( z = C -> ( B F z ) = ( B F C ) )
14	13	oveq2d 5983	. . . 4 |- ( z = C -> ( A F ( B F z ) ) = ( A F ( B F C ) ) )
15	12, 14	eqeq12d 2222	. . 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 2900	. 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 281	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 104   /\ w3a 981   = wceq 1373   e. wcel 2178   A.wral 2486  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  caovassd  6129  caovass  6130  seq3split  10670  seqsplitg  10671  seq3caopr  10677  seqcaoprg  10678  seqf1oglem2  10702  grpinvalem  13332  grpinva  13333  grprida  13334