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Theorem caovassg 6372
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 2804 . 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 6203 . . . . 5  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43oveq1d 6211 . . . 4  |-  ( x  =  A  ->  (
( x F y ) F z )  =  ( ( A F y ) F z ) )
5 oveq1 6203 . . . 4  |-  ( x  =  A  ->  (
x F ( y F z ) )  =  ( A F ( y F z ) ) )
64, 5eqeq12d 2404 . . 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 6204 . . . . 5  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87oveq1d 6211 . . . 4  |-  ( y  =  B  ->  (
( A F y ) F z )  =  ( ( A F B ) F z ) )
9 oveq1 6203 . . . . 5  |-  ( y  =  B  ->  (
y F z )  =  ( B F z ) )
109oveq2d 6212 . . . 4  |-  ( y  =  B  ->  ( A F ( y F z ) )  =  ( A F ( B F z ) ) )
118, 10eqeq12d 2404 . . 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 6204 . . . 4  |-  ( z  =  C  ->  (
( A F B ) F z )  =  ( ( A F B ) F C ) )
13 oveq2 6204 . . . . 5  |-  ( z  =  C  ->  ( B F z )  =  ( B F C ) )
1413oveq2d 6212 . . . 4  |-  ( z  =  C  ->  ( A F ( B F z ) )  =  ( A F ( B F C ) ) )
1512, 14eqeq12d 2404 . . 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 3147 . 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 467 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732  (class class class)co 6196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-iota 5460  df-fv 5504  df-ov 6199
This theorem is referenced by:  caovassd  6373  caovass  6374  grprinvlem  6412  grprinvd  6413  grpridd  6414  seqsplit  12043  seqcaopr  12047  seqf1olem2  12050
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