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Theorem caovdird 5704
Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovdirg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K ) )  -> 
( ( x F y ) G z )  =  ( ( x G z ) H ( y G z ) ) )
caovdird.2  |-  ( ph  ->  A  e.  S )
caovdird.3  |-  ( ph  ->  B  e.  S )
caovdird.4  |-  ( ph  ->  C  e.  K )
Assertion
Ref Expression
caovdird  |-  ( ph  ->  ( ( A F B ) G C )  =  ( ( A G C ) H ( 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, H, y, z   
x, K, y, z   
x, S, y, z

Proof of Theorem caovdird
StepHypRef Expression
1 id 19 . 2  |-  ( ph  ->  ph )
2 caovdird.2 . 2  |-  ( ph  ->  A  e.  S )
3 caovdird.3 . 2  |-  ( ph  ->  B  e.  S )
4 caovdird.4 . 2  |-  ( ph  ->  C  e.  K )
5 caovdirg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K ) )  -> 
( ( x F y ) G z )  =  ( ( x G z ) H ( y G z ) ) )
65caovdirg 5703 . 2  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  K ) )  -> 
( ( A F B ) G C )  =  ( ( A G C ) H ( B G C ) ) )
71, 2, 3, 4, 6syl13anc 1172 1  |-  ( ph  ->  ( ( A F B ) G C )  =  ( ( A G C ) H ( 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 5537
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 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-iota 4891  df-fv 4934  df-ov 5540
This theorem is referenced by:  caovdilemd  5717  recexgt0sr  7001
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