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Theorem caovdirg 5941
Description: Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)
Hypothesis
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 ) ) )
Assertion
Ref Expression
caovdirg  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  K ) )  -> 
( ( 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 caovdirg
StepHypRef Expression
1 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 ) ) )
21ralrimivvva 2513 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  A. z  e.  K  ( ( x F y ) G z )  =  ( ( x G z ) H ( y G z ) ) )
3 oveq1 5774 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43oveq1d 5782 . . . 4  |-  ( x  =  A  ->  (
( x F y ) G z )  =  ( ( A F y ) G z ) )
5 oveq1 5774 . . . . 5  |-  ( x  =  A  ->  (
x G z )  =  ( A G z ) )
65oveq1d 5782 . . . 4  |-  ( x  =  A  ->  (
( x G z ) H ( y G z ) )  =  ( ( A G z ) H ( y G z ) ) )
74, 6eqeq12d 2152 . . 3  |-  ( x  =  A  ->  (
( ( x F y ) G z )  =  ( ( x G z ) H ( y G z ) )  <->  ( ( A F y ) G z )  =  ( ( A G z ) H ( y G z ) ) ) )
8 oveq2 5775 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
98oveq1d 5782 . . . 4  |-  ( y  =  B  ->  (
( A F y ) G z )  =  ( ( A F B ) G z ) )
10 oveq1 5774 . . . . 5  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1110oveq2d 5783 . . . 4  |-  ( y  =  B  ->  (
( A G z ) H ( y G z ) )  =  ( ( A G z ) H ( B G z ) ) )
129, 11eqeq12d 2152 . . 3  |-  ( y  =  B  ->  (
( ( A F y ) G z )  =  ( ( A G z ) H ( y G z ) )  <->  ( ( A F B ) G z )  =  ( ( A G z ) H ( B G z ) ) ) )
13 oveq2 5775 . . . 4  |-  ( z  =  C  ->  (
( A F B ) G z )  =  ( ( A F B ) G C ) )
14 oveq2 5775 . . . . 5  |-  ( z  =  C  ->  ( A G z )  =  ( A G C ) )
15 oveq2 5775 . . . . 5  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
1614, 15oveq12d 5785 . . . 4  |-  ( z  =  C  ->  (
( A G z ) H ( B G z ) )  =  ( ( A G C ) H ( B G C ) ) )
1713, 16eqeq12d 2152 . . 3  |-  ( z  =  C  ->  (
( ( A F B ) G z )  =  ( ( A G z ) H ( B G z ) )  <->  ( ( A F B ) G C )  =  ( ( A G C ) H ( B G C ) ) ) )
187, 12, 17rspc3v 2800 . 2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  K )  ->  ( A. x  e.  S  A. y  e.  S  A. z  e.  K  ( ( x F y ) G z )  =  ( ( x G z ) H ( y G z ) )  ->  ( ( A F B ) G C )  =  ( ( A G C ) H ( B G C ) ) ) )
192, 18mpan9 279 1  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  K ) )  -> 
( ( A F B ) G C )  =  ( ( A G C ) H ( B G C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   A.wral 2414  (class class class)co 5767
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 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770
This theorem is referenced by:  caovdird  5942  caovlem2d  5956
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