Theorem caovdir2d 5708

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 5707	. 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 5685	. . 3 |- ( ph -> ( A F B ) e. S )
9	6, 8, 2	caovcomd 5688	. 2 |- ( ph -> ( ( A F B ) G C ) = ( C G ( A F B ) ) )
10	6, 3, 2	caovcomd 5688	. . 3 |- ( ph -> ( A G C ) = ( C G A ) )
11	6, 4, 2	caovcomd 5688	. . 3 |- ( ph -> ( B G C ) = ( C G B ) )
12	10, 11	oveq12d 5561	. 2 |- ( ph -> ( ( A G C ) F ( B G C ) ) = ( ( C G A ) F ( C G B ) ) )
13	5, 9, 12	3eqtr4d 2124	1 |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   = wceq 1285   e. wcel 1434  (class class class)co 5543

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546

This theorem is referenced by:  addcmpblnq  6619  ltanqg  6652  addcmpblnq0  6695  mulasssrg  6997  mulgt0sr  7016  mulextsr1lem  7018