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Theorem caovdir2d 6255
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
StepHypRef 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 )
51, 2, 3, 4caovdid 6254 . 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 )
87, 3, 4caovcld 6232 . . 3  |-  ( ph  ->  ( A F B )  e.  S )
96, 8, 2caovcomd 6235 . 2  |-  ( ph  ->  ( ( A F B ) G C )  =  ( C G ( A F B ) ) )
106, 3, 2caovcomd 6235 . . 3  |-  ( ph  ->  ( A G C )  =  ( C G A ) )
116, 4, 2caovcomd 6235 . . 3  |-  ( ph  ->  ( B G C )  =  ( C G B ) )
1210, 11oveq12d 6091 . 2  |-  ( ph  ->  ( ( A G C ) F ( B G C ) )  =  ( ( C G A ) F ( C G B ) ) )
135, 9, 123eqtr4d 2477 1  |-  ( ph  ->  ( ( A F B ) G C )  =  ( ( A G C ) F ( B G C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725  (class class class)co 6073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
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