Theorem caovdir2d 5955

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 5954	. 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 5932	. . 3 |- ( ph -> ( A F B ) e. S )
9	6, 8, 2	caovcomd 5935	. 2 |- ( ph -> ( ( A F B ) G C ) = ( C G ( A F B ) ) )
10	6, 3, 2	caovcomd 5935	. . 3 |- ( ph -> ( A G C ) = ( C G A ) )
11	6, 4, 2	caovcomd 5935	. . 3 |- ( ph -> ( B G C ) = ( C G B ) )
12	10, 11	oveq12d 5800	. 2 |- ( ph -> ( ( A G C ) F ( B G C ) ) = ( ( C G A ) F ( C G B ) ) )
13	5, 9, 12	3eqtr4d 2183	1 |- ( ph -> ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1332   e. wcel 1481  (class class class)co 5782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by:  addcmpblnq  7199  ltanqg  7232  addcmpblnq0  7275  mulasssrg  7590  mulgt0sr  7610  mulextsr1lem  7612