Theorem caovdig 6051

Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)

Hypothesis

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

Assertion

Ref	Expression
caovdig	|- ( ( ph /\ ( A e. K /\ B e. S /\ C e. S ) ) -> ( A G ( B F C ) ) = ( ( A G B ) H ( A 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 caovdig

Step	Hyp	Ref	Expression
1		caovdig.1	. . 3 |- ( ( ph /\ ( x e. K /\ y e. S /\ z e. S ) ) -> ( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) )
2	1	ralrimivvva 2560	. 2 |- ( ph -> A. x e. K A. y e. S A. z e. S ( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) )
3		oveq1 5884	. . . 4 |- ( x = A -> ( x G ( y F z ) ) = ( A G ( y F z ) ) )
4		oveq1 5884	. . . . 5 |- ( x = A -> ( x G y ) = ( A G y ) )
5		oveq1 5884	. . . . 5 |- ( x = A -> ( x G z ) = ( A G z ) )
6	4, 5	oveq12d 5895	. . . 4 |- ( x = A -> ( ( x G y ) H ( x G z ) ) = ( ( A G y ) H ( A G z ) ) )
7	3, 6	eqeq12d 2192	. . 3 |- ( x = A -> ( ( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) <-> ( A G ( y F z ) ) = ( ( A G y ) H ( A G z ) ) ) )
8		oveq1 5884	. . . . 5 |- ( y = B -> ( y F z ) = ( B F z ) )
9	8	oveq2d 5893	. . . 4 |- ( y = B -> ( A G ( y F z ) ) = ( A G ( B F z ) ) )
10		oveq2 5885	. . . . 5 |- ( y = B -> ( A G y ) = ( A G B ) )
11	10	oveq1d 5892	. . . 4 |- ( y = B -> ( ( A G y ) H ( A G z ) ) = ( ( A G B ) H ( A G z ) ) )
12	9, 11	eqeq12d 2192	. . 3 |- ( y = B -> ( ( A G ( y F z ) ) = ( ( A G y ) H ( A G z ) ) <-> ( A G ( B F z ) ) = ( ( A G B ) H ( A G z ) ) ) )
13		oveq2 5885	. . . . 5 |- ( z = C -> ( B F z ) = ( B F C ) )
14	13	oveq2d 5893	. . . 4 |- ( z = C -> ( A G ( B F z ) ) = ( A G ( B F C ) ) )
15		oveq2 5885	. . . . 5 |- ( z = C -> ( A G z ) = ( A G C ) )
16	15	oveq2d 5893	. . . 4 |- ( z = C -> ( ( A G B ) H ( A G z ) ) = ( ( A G B ) H ( A G C ) ) )
17	14, 16	eqeq12d 2192	. . 3 |- ( z = C -> ( ( A G ( B F z ) ) = ( ( A G B ) H ( A G z ) ) <-> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) ) )
18	7, 12, 17	rspc3v 2859	. 2 |- ( ( A e. K /\ B e. S /\ C e. S ) -> ( A. x e. K A. y e. S A. z e. S ( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) ) )
19	2, 18	mpan9 281	1 |- ( ( ph /\ ( A e. K /\ B e. S /\ C e. S ) ) -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455  (class class class)co 5877

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by:  caovdid  6052  caovdi  6056  srgdilem  13157  ringdilem  13200