Theorem caovdir2d 6054

Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovdir2d.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) )
caovdir2d.2	|- ( ph -> A e. S )
caovdir2d.3	|- ( ph -> B e. S )
caovdir2d.4	|- ( ph -> C e. S )
caovdir2d.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
caovdir2d.com	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) = ( y G x ) )

Assertion

Ref	Expression
caovdir2d	|- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) F ( B G 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, G, y, z   x, S, y, z

Proof of Theorem caovdir2d

Step	Hyp	Ref	Expression
1		caovdir2d.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) )
2		caovdir2d.4	. . 3 |- ( ph -> C e. S )
3		caovdir2d.2	. . 3 |- ( ph -> A e. S )
4		caovdir2d.3	. . 3 |- ( ph -> B e. S )
5	1, 2, 3, 4	caovdid 6053	. 2 |- ( ph -> ( C G ( A F B ) ) = ( ( C G A ) F ( C G B ) ) )
6		caovdir2d.com	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) = ( y G x ) )
7		caovdir2d.cl	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
8	7, 3, 4	caovcld 6031	. . 3 |- ( ph -> ( A F B ) e. S )
9	6, 8, 2	caovcomd 6034	. 2 |- ( ph -> ( ( A F B ) G C ) = ( C G ( A F B ) ) )
10	6, 3, 2	caovcomd 6034	. . 3 |- ( ph -> ( A G C ) = ( C G A ) )
11	6, 4, 2	caovcomd 6034	. . 3 |- ( ph -> ( B G C ) = ( C G B ) )
12	10, 11	oveq12d 5896	. 2 |- ( ph -> ( ( A G C ) F ( B G C ) ) = ( ( C G A ) F ( C G B ) ) )
13	5, 9, 12	3eqtr4d 2220	1 |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148  (class class class)co 5878

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5881

This theorem is referenced by:  addcmpblnq  7369  ltanqg  7402  addcmpblnq0  7445  mulasssrg  7760  mulgt0sr  7780  mulextsr1lem  7782