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Theorem caovdir2d 5708
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 5707 . 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 5685 . . 3  |-  ( ph  ->  ( A F B )  e.  S )
96, 8, 2caovcomd 5688 . 2  |-  ( ph  ->  ( ( A F B ) G C )  =  ( C G ( A F B ) ) )
106, 3, 2caovcomd 5688 . . 3  |-  ( ph  ->  ( A G C )  =  ( C G A ) )
116, 4, 2caovcomd 5688 . . 3  |-  ( ph  ->  ( B G C )  =  ( C G B ) )
1210, 11oveq12d 5561 . 2  |-  ( ph  ->  ( ( A G C ) F ( B G C ) )  =  ( ( C G A ) F ( C G B ) ) )
135, 9, 123eqtr4d 2124 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 102    /\ w3a 920    = wceq 1285    e. wcel 1434  (class class class)co 5543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546
This theorem is referenced by:  addcmpblnq  6619  ltanqg  6652  addcmpblnq0  6695  mulasssrg  6997  mulgt0sr  7016  mulextsr1lem  7018
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