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Theorem caovdirg 6037
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 2636 . 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 5865 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43oveq1d 5873 . . . 4  |-  ( x  =  A  ->  (
( x F y ) G z )  =  ( ( A F y ) G z ) )
5 oveq1 5865 . . . . 5  |-  ( x  =  A  ->  (
x G z )  =  ( A G z ) )
65oveq1d 5873 . . . 4  |-  ( x  =  A  ->  (
( x G z ) H ( y G z ) )  =  ( ( A G z ) H ( y G z ) ) )
74, 6eqeq12d 2297 . . 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 5866 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
98oveq1d 5873 . . . 4  |-  ( y  =  B  ->  (
( A F y ) G z )  =  ( ( A F B ) G z ) )
10 oveq1 5865 . . . . 5  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1110oveq2d 5874 . . . 4  |-  ( y  =  B  ->  (
( A G z ) H ( y G z ) )  =  ( ( A G z ) H ( B G z ) ) )
129, 11eqeq12d 2297 . . 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 5866 . . . 4  |-  ( z  =  C  ->  (
( A F B ) G z )  =  ( ( A F B ) G C ) )
14 oveq2 5866 . . . . 5  |-  ( z  =  C  ->  ( A G z )  =  ( A G C ) )
15 oveq2 5866 . . . . 5  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
1614, 15oveq12d 5876 . . . 4  |-  ( z  =  C  ->  (
( A G z ) H ( B G z ) )  =  ( ( A G C ) H ( B G C ) ) )
1713, 16eqeq12d 2297 . . 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 2893 . 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 455 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543  (class class class)co 5858
This theorem is referenced by:  caovdird  6038  rngi  15353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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