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Theorem caovdirg 6267
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 2801 . 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 6091 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43oveq1d 6099 . . . 4  |-  ( x  =  A  ->  (
( x F y ) G z )  =  ( ( A F y ) G z ) )
5 oveq1 6091 . . . . 5  |-  ( x  =  A  ->  (
x G z )  =  ( A G z ) )
65oveq1d 6099 . . . 4  |-  ( x  =  A  ->  (
( x G z ) H ( y G z ) )  =  ( ( A G z ) H ( y G z ) ) )
74, 6eqeq12d 2452 . . 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 6092 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
98oveq1d 6099 . . . 4  |-  ( y  =  B  ->  (
( A F y ) G z )  =  ( ( A F B ) G z ) )
10 oveq1 6091 . . . . 5  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1110oveq2d 6100 . . . 4  |-  ( y  =  B  ->  (
( A G z ) H ( y G z ) )  =  ( ( A G z ) H ( B G z ) ) )
129, 11eqeq12d 2452 . . 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 6092 . . . 4  |-  ( z  =  C  ->  (
( A F B ) G z )  =  ( ( A F B ) G C ) )
14 oveq2 6092 . . . . 5  |-  ( z  =  C  ->  ( A G z )  =  ( A G C ) )
15 oveq2 6092 . . . . 5  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
1614, 15oveq12d 6102 . . . 4  |-  ( z  =  C  ->  (
( A G z ) H ( B G z ) )  =  ( ( A G C ) H ( B G C ) ) )
1713, 16eqeq12d 2452 . . 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 3063 . 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 457 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707  (class class class)co 6084
This theorem is referenced by:  caovdird  6268  rngi  15681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087
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