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Theorem caov4d 5921
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovd.1  |-  ( ph  ->  A  e.  S )
caovd.2  |-  ( ph  ->  B  e.  S )
caovd.3  |-  ( ph  ->  C  e.  S )
caovd.com  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
caovd.ass  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
caovd.4  |-  ( ph  ->  D  e.  S )
caovd.cl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
Assertion
Ref Expression
caov4d  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    x, D, y, z    ph, x, y, z   
x, F, y, z   
x, S, y, z

Proof of Theorem caov4d
StepHypRef Expression
1 caovd.2 . . . 4  |-  ( ph  ->  B  e.  S )
2 caovd.3 . . . 4  |-  ( ph  ->  C  e.  S )
3 caovd.4 . . . 4  |-  ( ph  ->  D  e.  S )
4 caovd.com . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
5 caovd.ass . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
61, 2, 3, 4, 5caov12d 5918 . . 3  |-  ( ph  ->  ( B F ( C F D ) )  =  ( C F ( B F D ) ) )
76oveq2d 5756 . 2  |-  ( ph  ->  ( A F ( B F ( C F D ) ) )  =  ( A F ( C F ( B F D ) ) ) )
8 caovd.1 . . 3  |-  ( ph  ->  A  e.  S )
9 caovd.cl . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
109, 2, 3caovcld 5890 . . 3  |-  ( ph  ->  ( C F D )  e.  S )
115, 8, 1, 10caovassd 5896 . 2  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( A F ( B F ( C F D ) ) ) )
129, 1, 3caovcld 5890 . . 3  |-  ( ph  ->  ( B F D )  e.  S )
135, 8, 2, 12caovassd 5896 . 2  |-  ( ph  ->  ( ( A F C ) F ( B F D ) )  =  ( A F ( C F ( B F D ) ) ) )
147, 11, 133eqtr4d 2158 1  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( A F C ) F ( B F D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314    e. wcel 1463  (class class class)co 5740
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-iota 5056  df-fv 5099  df-ov 5743
This theorem is referenced by:  caov411d  5922  caov42d  5923  ecopovtrn  6492  ecopovtrng  6495  addcmpblnq  7139  mulcmpblnq  7140  ordpipqqs  7146  distrnqg  7159  ltsonq  7170  ltanqg  7172  ltmnqg  7173  addcmpblnq0  7215  mulcmpblnq0  7216  distrnq0  7231  prarloclemlo  7266  addlocprlemeqgt  7304  addcanprleml  7386  recexprlem1ssl  7405  recexprlem1ssu  7406  mulcmpblnrlemg  7512  distrsrg  7531  ltasrg  7542  mulgt0sr  7550  prsradd  7558  axdistr  7646
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