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Theorem caovdir2d 5947
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 5946 . 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 5924 . . 3  |-  ( ph  ->  ( A F B )  e.  S )
96, 8, 2caovcomd 5927 . 2  |-  ( ph  ->  ( ( A F B ) G C )  =  ( C G ( A F B ) ) )
106, 3, 2caovcomd 5927 . . 3  |-  ( ph  ->  ( A G C )  =  ( C G A ) )
116, 4, 2caovcomd 5927 . . 3  |-  ( ph  ->  ( B G C )  =  ( C G B ) )
1210, 11oveq12d 5792 . 2  |-  ( ph  ->  ( ( A G C ) F ( B G C ) )  =  ( ( C G A ) F ( C G B ) ) )
135, 9, 123eqtr4d 2182 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 103    /\ w3a 962    = wceq 1331    e. wcel 1480  (class class class)co 5774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  addcmpblnq  7182  ltanqg  7215  addcmpblnq0  7258  mulasssrg  7573  mulgt0sr  7593  mulextsr1lem  7595
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