Theorem caovdir2d 6136

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 6135	. 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 6113	. . 3 |- ( ph -> ( A F B ) e. S )
9	6, 8, 2	caovcomd 6116	. 2 |- ( ph -> ( ( A F B ) G C ) = ( C G ( A F B ) ) )
10	6, 3, 2	caovcomd 6116	. . 3 |- ( ph -> ( A G C ) = ( C G A ) )
11	6, 4, 2	caovcomd 6116	. . 3 |- ( ph -> ( B G C ) = ( C G B ) )
12	10, 11	oveq12d 5975	. 2 |- ( ph -> ( ( A G C ) F ( B G C ) ) = ( ( C G A ) F ( C G B ) ) )
13	5, 9, 12	3eqtr4d 2249	1 |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2177  (class class class)co 5957

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 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-iota 5241  df-fv 5288  df-ov 5960

This theorem is referenced by:  addcmpblnq  7500  ltanqg  7533  addcmpblnq0  7576  mulasssrg  7891  mulgt0sr  7911  mulextsr1lem  7913