Theorem caovdir2d

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

	Ref	Expression
Hypotheses	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	caovdir2d	|- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) )

Proof

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