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Theorem caovdirg 6227
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 2763 . 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 6051 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43oveq1d 6059 . . . 4  |-  ( x  =  A  ->  (
( x F y ) G z )  =  ( ( A F y ) G z ) )
5 oveq1 6051 . . . . 5  |-  ( x  =  A  ->  (
x G z )  =  ( A G z ) )
65oveq1d 6059 . . . 4  |-  ( x  =  A  ->  (
( x G z ) H ( y G z ) )  =  ( ( A G z ) H ( y G z ) ) )
74, 6eqeq12d 2422 . . 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 6052 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
98oveq1d 6059 . . . 4  |-  ( y  =  B  ->  (
( A F y ) G z )  =  ( ( A F B ) G z ) )
10 oveq1 6051 . . . . 5  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1110oveq2d 6060 . . . 4  |-  ( y  =  B  ->  (
( A G z ) H ( y G z ) )  =  ( ( A G z ) H ( B G z ) ) )
129, 11eqeq12d 2422 . . 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 6052 . . . 4  |-  ( z  =  C  ->  (
( A F B ) G z )  =  ( ( A F B ) G C ) )
14 oveq2 6052 . . . . 5  |-  ( z  =  C  ->  ( A G z )  =  ( A G C ) )
15 oveq2 6052 . . . . 5  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
1614, 15oveq12d 6062 . . . 4  |-  ( z  =  C  ->  (
( A G z ) H ( B G z ) )  =  ( ( A G C ) H ( B G C ) ) )
1713, 16eqeq12d 2422 . . 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 3025 . 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 456 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670  (class class class)co 6044
This theorem is referenced by:  caovdird  6228  rngi  15635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-iota 5381  df-fv 5425  df-ov 6047
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