Theorem caovdirg 5678

Description: Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)

Hypothesis

Ref	Expression
caovdirg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. K ) ) -> ( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) )

Assertion

Ref	Expression
caovdirg	|- ( ( ph /\ ( A e. S /\ B e. S /\ C e. K ) ) -> ( ( A F B ) G C ) = ( ( A G C ) H ( 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, H, y, z   x, K, y, z   x, S, y, z

Proof of Theorem caovdirg

Step	Hyp	Ref	Expression
1		caovdirg.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. K ) ) -> ( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) )
2	1	ralrimivvva 2402	. 2 |- ( ph -> A. x e. S A. y e. S A. z e. K ( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) )
3		oveq1 5519	. . . . 5 |- ( x = A -> ( x F y ) = ( A F y ) )
4	3	oveq1d 5527	. . . 4 |- ( x = A -> ( ( x F y ) G z ) = ( ( A F y ) G z ) )
5		oveq1 5519	. . . . 5 |- ( x = A -> ( x G z ) = ( A G z ) )
6	5	oveq1d 5527	. . . 4 |- ( x = A -> ( ( x G z ) H ( y G z ) ) = ( ( A G z ) H ( y G z ) ) )
7	4, 6	eqeq12d 2054	. . 3 |- ( x = A -> ( ( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) <-> ( ( A F y ) G z ) = ( ( A G z ) H ( y G z ) ) ) )
8		oveq2 5520	. . . . 5 |- ( y = B -> ( A F y ) = ( A F B ) )
9	8	oveq1d 5527	. . . 4 |- ( y = B -> ( ( A F y ) G z ) = ( ( A F B ) G z ) )
10		oveq1 5519	. . . . 5 |- ( y = B -> ( y G z ) = ( B G z ) )
11	10	oveq2d 5528	. . . 4 |- ( y = B -> ( ( A G z ) H ( y G z ) ) = ( ( A G z ) H ( B G z ) ) )
12	9, 11	eqeq12d 2054	. . 3 |- ( y = B -> ( ( ( A F y ) G z ) = ( ( A G z ) H ( y G z ) ) <-> ( ( A F B ) G z ) = ( ( A G z ) H ( B G z ) ) ) )
13		oveq2 5520	. . . 4 |- ( z = C -> ( ( A F B ) G z ) = ( ( A F B ) G C ) )
14		oveq2 5520	. . . . 5 |- ( z = C -> ( A G z ) = ( A G C ) )
15		oveq2 5520	. . . . 5 |- ( z = C -> ( B G z ) = ( B G C ) )
16	14, 15	oveq12d 5530	. . . 4 |- ( z = C -> ( ( A G z ) H ( B G z ) ) = ( ( A G C ) H ( B G C ) ) )
17	13, 16	eqeq12d 2054	. . 3 |- ( z = C -> ( ( ( A F B ) G z ) = ( ( A G z ) H ( B G z ) ) <-> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) ) )
18	7, 12, 17	rspc3v 2665	. 2 |- ( ( A e. S /\ B e. S /\ C e. K ) -> ( A. x e. S A. y e. S A. z e. K ( ( x F y ) G z ) = ( ( x G z ) H ( y G z ) ) -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) ) )
19	2, 18	mpan9 265	1 |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. K ) ) -> ( ( A F B ) G C ) = ( ( A G C ) H ( B G C ) ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393   A.wral 2306  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  caovdird  5679  caovlem2d  5693