Theorem caovdirg 6182

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 2613	. 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 6007	. . . . 5 |- ( x = A -> ( x F y ) = ( A F y ) )
4	3	oveq1d 6015	. . . 4 |- ( x = A -> ( ( x F y ) G z ) = ( ( A F y ) G z ) )
5		oveq1 6007	. . . . 5 |- ( x = A -> ( x G z ) = ( A G z ) )
6	5	oveq1d 6015	. . . 4 |- ( x = A -> ( ( x G z ) H ( y G z ) ) = ( ( A G z ) H ( y G z ) ) )
7	4, 6	eqeq12d 2244	. . 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 6008	. . . . 5 |- ( y = B -> ( A F y ) = ( A F B ) )
9	8	oveq1d 6015	. . . 4 |- ( y = B -> ( ( A F y ) G z ) = ( ( A F B ) G z ) )
10		oveq1 6007	. . . . 5 |- ( y = B -> ( y G z ) = ( B G z ) )
11	10	oveq2d 6016	. . . 4 |- ( y = B -> ( ( A G z ) H ( y G z ) ) = ( ( A G z ) H ( B G z ) ) )
12	9, 11	eqeq12d 2244	. . 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 6008	. . . 4 |- ( z = C -> ( ( A F B ) G z ) = ( ( A F B ) G C ) )
14		oveq2 6008	. . . . 5 |- ( z = C -> ( A G z ) = ( A G C ) )
15		oveq2 6008	. . . . 5 |- ( z = C -> ( B G z ) = ( B G C ) )
16	14, 15	oveq12d 6018	. . . 4 |- ( z = C -> ( ( A G z ) H ( B G z ) ) = ( ( A G C ) H ( B G C ) ) )
17	13, 16	eqeq12d 2244	. . 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 2923	. 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 281	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 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508  (class class class)co 6000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003

This theorem is referenced by:  caovdird  6183  caovlem2d  6197  srgdilem  13927  ringdilem  13970