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Theorem caovassg 6245
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 2799 . 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 6088 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43oveq1d 6096 . . . 4  |-  ( x  =  A  ->  (
( x F y ) F z )  =  ( ( A F y ) F z ) )
5 oveq1 6088 . . . 4  |-  ( x  =  A  ->  (
x F ( y F z ) )  =  ( A F ( y F z ) ) )
64, 5eqeq12d 2450 . . 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 6089 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87oveq1d 6096 . . . 4  |-  ( y  =  B  ->  (
( A F y ) F z )  =  ( ( A F B ) F z ) )
9 oveq1 6088 . . . . 5  |-  ( y  =  B  ->  (
y F z )  =  ( B F z ) )
109oveq2d 6097 . . . 4  |-  ( y  =  B  ->  ( A F ( y F z ) )  =  ( A F ( B F z ) ) )
118, 10eqeq12d 2450 . . 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 6089 . . . 4  |-  ( z  =  C  ->  (
( A F B ) F z )  =  ( ( A F B ) F C ) )
13 oveq2 6089 . . . . 5  |-  ( z  =  C  ->  ( B F z )  =  ( B F C ) )
1413oveq2d 6097 . . . 4  |-  ( z  =  C  ->  ( A F ( B F z ) )  =  ( A F ( B F C ) ) )
1512, 14eqeq12d 2450 . . 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 3061 . 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 456 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705  (class class class)co 6081
This theorem is referenced by:  caovassd  6246  caovass  6247  grprinvlem  6285  grprinvd  6286  grpridd  6287  seqsplit  11356  seqcaopr  11360  seqf1olem2  11363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
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