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Theorem caovdig 5706
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 2445 . 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 5550 . . . 4  |-  ( x  =  A  ->  (
x G ( y F z ) )  =  ( A G ( y F z ) ) )
4 oveq1 5550 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
5 oveq1 5550 . . . . 5  |-  ( x  =  A  ->  (
x G z )  =  ( A G z ) )
64, 5oveq12d 5561 . . . 4  |-  ( x  =  A  ->  (
( x G y ) H ( x G z ) )  =  ( ( A G y ) H ( A G z ) ) )
73, 6eqeq12d 2096 . . 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 5550 . . . . 5  |-  ( y  =  B  ->  (
y F z )  =  ( B F z ) )
98oveq2d 5559 . . . 4  |-  ( y  =  B  ->  ( A G ( y F z ) )  =  ( A G ( B F z ) ) )
10 oveq2 5551 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1110oveq1d 5558 . . . 4  |-  ( y  =  B  ->  (
( A G y ) H ( A G z ) )  =  ( ( A G B ) H ( A G z ) ) )
129, 11eqeq12d 2096 . . 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 5551 . . . . 5  |-  ( z  =  C  ->  ( B F z )  =  ( B F C ) )
1413oveq2d 5559 . . . 4  |-  ( z  =  C  ->  ( A G ( B F z ) )  =  ( A G ( B F C ) ) )
15 oveq2 5551 . . . . 5  |-  ( z  =  C  ->  ( A G z )  =  ( A G C ) )
1615oveq2d 5559 . . . 4  |-  ( z  =  C  ->  (
( A G B ) H ( A G z ) )  =  ( ( A G B ) H ( A G C ) ) )
1714, 16eqeq12d 2096 . . 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 2717 . 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 275 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 102    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2349  (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:  caovdid  5707  caovdi  5711
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