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Metamath Proof Explorer

Theorem caovdig

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

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

Proof

Step Hyp Ref Expression
1 caovdig.1 
 |-  ( ( 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 
 |-  ( 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 
 |-  ( x = A -> ( x G ( y F z ) ) = ( A G ( y F z ) ) )
4 oveq1 
 |-  ( x = A -> ( x G y ) = ( A G y ) )
5 oveq1 
 |-  ( x = A -> ( x G z ) = ( A G z ) )
6 4 5 oveq12d 
 |-  ( x = A -> ( ( x G y ) H ( x G z ) ) = ( ( A G y ) H ( A G z ) ) )
7 3 6 eqeq12d 
 |-  ( 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 
 |-  ( y = B -> ( y F z ) = ( B F z ) )
9 8 oveq2d 
 |-  ( y = B -> ( A G ( y F z ) ) = ( A G ( B F z ) ) )
10 oveq2 
 |-  ( y = B -> ( A G y ) = ( A G B ) )
11 10 oveq1d 
 |-  ( y = B -> ( ( A G y ) H ( A G z ) ) = ( ( A G B ) H ( A G z ) ) )
12 9 11 eqeq12d 
 |-  ( 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 
 |-  ( z = C -> ( B F z ) = ( B F C ) )
14 13 oveq2d 
 |-  ( z = C -> ( A G ( B F z ) ) = ( A G ( B F C ) ) )
15 oveq2 
 |-  ( z = C -> ( A G z ) = ( A G C ) )
16 15 oveq2d 
 |-  ( z = C -> ( ( A G B ) H ( A G z ) ) = ( ( A G B ) H ( A G C ) ) )
17 14 16 eqeq12d 
 |-  ( 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 
 |-  ( ( 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 
 |-  ( ( ph /\ ( A e. K /\ B e. S /\ C e. S ) ) -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) )