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Theorem caovdir2d 6036
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 6035 . 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 6013 . . 3  |-  ( ph  ->  ( A F B )  e.  S )
96, 8, 2caovcomd 6016 . 2  |-  ( ph  ->  ( ( A F B ) G C )  =  ( C G ( A F B ) ) )
106, 3, 2caovcomd 6016 . . 3  |-  ( ph  ->  ( A G C )  =  ( C G A ) )
116, 4, 2caovcomd 6016 . . 3  |-  ( ph  ->  ( B G C )  =  ( C G B ) )
1210, 11oveq12d 5876 . 2  |-  ( ph  ->  ( ( A G C ) F ( B G C ) )  =  ( ( C G A ) F ( C G B ) ) )
135, 9, 123eqtr4d 2325 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684  (class class class)co 5858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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