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Theorem caovdig 5913
Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
caovdig.1  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )
Assertion
Ref Expression
caovdig  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  S  /\  C  e.  S ) )  -> 
( A G ( B F C ) )  =  ( ( A G B ) H ( A 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 caovdig
StepHypRef Expression
1 caovdig.1 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )
21ralrimivvva 2492 . 2  |-  ( ph  ->  A. x  e.  K  A. y  e.  S  A. z  e.  S  ( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )
3 oveq1 5749 . . . 4  |-  ( x  =  A  ->  (
x G ( y F z ) )  =  ( A G ( y F z ) ) )
4 oveq1 5749 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
5 oveq1 5749 . . . . 5  |-  ( x  =  A  ->  (
x G z )  =  ( A G z ) )
64, 5oveq12d 5760 . . . 4  |-  ( x  =  A  ->  (
( x G y ) H ( x G z ) )  =  ( ( A G y ) H ( A G z ) ) )
73, 6eqeq12d 2132 . . 3  |-  ( x  =  A  ->  (
( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) )  <->  ( A G ( y F z ) )  =  ( ( A G y ) H ( A G z ) ) ) )
8 oveq1 5749 . . . . 5  |-  ( y  =  B  ->  (
y F z )  =  ( B F z ) )
98oveq2d 5758 . . . 4  |-  ( y  =  B  ->  ( A G ( y F z ) )  =  ( A G ( B F z ) ) )
10 oveq2 5750 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1110oveq1d 5757 . . . 4  |-  ( y  =  B  ->  (
( A G y ) H ( A G z ) )  =  ( ( A G B ) H ( A G z ) ) )
129, 11eqeq12d 2132 . . 3  |-  ( y  =  B  ->  (
( A G ( y F z ) )  =  ( ( A G y ) H ( A G z ) )  <->  ( A G ( B F z ) )  =  ( ( A G B ) H ( A G z ) ) ) )
13 oveq2 5750 . . . . 5  |-  ( z  =  C  ->  ( B F z )  =  ( B F C ) )
1413oveq2d 5758 . . . 4  |-  ( z  =  C  ->  ( A G ( B F z ) )  =  ( A G ( B F C ) ) )
15 oveq2 5750 . . . . 5  |-  ( z  =  C  ->  ( A G z )  =  ( A G C ) )
1615oveq2d 5758 . . . 4  |-  ( z  =  C  ->  (
( A G B ) H ( A G z ) )  =  ( ( A G B ) H ( A G C ) ) )
1714, 16eqeq12d 2132 . . 3  |-  ( z  =  C  ->  (
( A G ( B F z ) )  =  ( ( A G B ) H ( A G z ) )  <->  ( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) ) )
187, 12, 17rspc3v 2779 . 2  |-  ( ( A  e.  K  /\  B  e.  S  /\  C  e.  S )  ->  ( A. x  e.  K  A. y  e.  S  A. z  e.  S  ( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) )  ->  ( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) ) )
192, 18mpan9 279 1  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  S  /\  C  e.  S ) )  -> 
( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465   A.wral 2393  (class class class)co 5742
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  caovdid  5914  caovdi  5918
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