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Theorem caovassg 6034
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 2649 . 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 5881 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43oveq1d 5889 . . . 4  |-  ( x  =  A  ->  (
( x F y ) F z )  =  ( ( A F y ) F z ) )
5 oveq1 5881 . . . 4  |-  ( x  =  A  ->  (
x F ( y F z ) )  =  ( A F ( y F z ) ) )
64, 5eqeq12d 2310 . . 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 5882 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87oveq1d 5889 . . . 4  |-  ( y  =  B  ->  (
( A F y ) F z )  =  ( ( A F B ) F z ) )
9 oveq1 5881 . . . . 5  |-  ( y  =  B  ->  (
y F z )  =  ( B F z ) )
109oveq2d 5890 . . . 4  |-  ( y  =  B  ->  ( A F ( y F z ) )  =  ( A F ( B F z ) ) )
118, 10eqeq12d 2310 . . 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 5882 . . . 4  |-  ( z  =  C  ->  (
( A F B ) F z )  =  ( ( A F B ) F C ) )
13 oveq2 5882 . . . . 5  |-  ( z  =  C  ->  ( B F z )  =  ( B F C ) )
1413oveq2d 5890 . . . 4  |-  ( z  =  C  ->  ( A F ( B F z ) )  =  ( A F ( B F C ) ) )
1512, 14eqeq12d 2310 . . 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 2906 . 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 455 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556  (class class class)co 5874
This theorem is referenced by:  caovassd  6035  caovass  6036  grprinvlem  6074  grprinvd  6075  grpridd  6076  seqsplit  11095  seqcaopr  11099  seqf1olem2  11102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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