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Theorem caovassg 5992
Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)
Hypothesis
Ref Expression
caovassg.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
Assertion
Ref Expression
caovassg  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )  -> 
( ( A F B ) F C )  =  ( A F ( B F 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, S, y, z

Proof of Theorem caovassg
StepHypRef Expression
1 caovassg.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
21ralrimivvva 2547 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  A. z  e.  S  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )
3 oveq1 5844 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43oveq1d 5852 . . . 4  |-  ( x  =  A  ->  (
( x F y ) F z )  =  ( ( A F y ) F z ) )
5 oveq1 5844 . . . 4  |-  ( x  =  A  ->  (
x F ( y F z ) )  =  ( A F ( y F z ) ) )
64, 5eqeq12d 2179 . . 3  |-  ( x  =  A  ->  (
( ( x F y ) F z )  =  ( x F ( y F z ) )  <->  ( ( A F y ) F z )  =  ( A F ( y F z ) ) ) )
7 oveq2 5845 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87oveq1d 5852 . . . 4  |-  ( y  =  B  ->  (
( A F y ) F z )  =  ( ( A F B ) F z ) )
9 oveq1 5844 . . . . 5  |-  ( y  =  B  ->  (
y F z )  =  ( B F z ) )
109oveq2d 5853 . . . 4  |-  ( y  =  B  ->  ( A F ( y F z ) )  =  ( A F ( B F z ) ) )
118, 10eqeq12d 2179 . . 3  |-  ( y  =  B  ->  (
( ( A F y ) F z )  =  ( A F ( y F z ) )  <->  ( ( A F B ) F z )  =  ( A F ( B F z ) ) ) )
12 oveq2 5845 . . . 4  |-  ( z  =  C  ->  (
( A F B ) F z )  =  ( ( A F B ) F C ) )
13 oveq2 5845 . . . . 5  |-  ( z  =  C  ->  ( B F z )  =  ( B F C ) )
1413oveq2d 5853 . . . 4  |-  ( z  =  C  ->  ( A F ( B F z ) )  =  ( A F ( B F C ) ) )
1512, 14eqeq12d 2179 . . 3  |-  ( z  =  C  ->  (
( ( A F B ) F z )  =  ( A F ( B F z ) )  <->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) ) )
166, 11, 15rspc3v 2842 . 2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A. x  e.  S  A. y  e.  S  A. z  e.  S  ( ( x F y ) F z )  =  ( x F ( y F z ) )  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) ) )
172, 16mpan9 279 1  |-  ( (
ph  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )  -> 
( ( A F B ) F C )  =  ( A F ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 967    = wceq 1342    e. wcel 2135   A.wral 2442  (class class class)co 5837
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-un 3116  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-iota 5148  df-fv 5191  df-ov 5840
This theorem is referenced by:  caovassd  5993  caovass  5994  grprinvlem  6028  grprinvd  6029  grpridd  6030  seq3split  10405  seq3caopr  10409
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