Theorem caovdir2d 6850

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 6849	. 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 6827	. . 3 |- ( ph -> ( A F B ) e. S )
9	6, 8, 2	caovcomd 6830	. 2 |- ( ph -> ( ( A F B ) G C ) = ( C G ( A F B ) ) )
10	6, 3, 2	caovcomd 6830	. . 3 |- ( ph -> ( A G C ) = ( C G A ) )
11	6, 4, 2	caovcomd 6830	. . 3 |- ( ph -> ( B G C ) = ( C G B ) )
12	10, 11	oveq12d 6668	. 2 |- ( ph -> ( ( A G C ) F ( B G C ) ) = ( ( C G A ) F ( C G B ) ) )
13	5, 9, 12	3eqtr4d 2666	1 |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)