MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caov411d Unicode version

Theorem caov411d 6172
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
caov411d  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( C F B ) F ( A 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 caov411d
StepHypRef Expression
1 caovd.2 . . 3  |-  ( ph  ->  B  e.  S )
2 caovd.1 . . 3  |-  ( ph  ->  A  e.  S )
3 caovd.3 . . 3  |-  ( ph  ->  C  e.  S )
4 caovd.com . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
5 caovd.ass . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
6 caovd.4 . . 3  |-  ( ph  ->  D  e.  S )
7 caovd.cl . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
81, 2, 3, 4, 5, 6, 7caov4d 6171 . 2  |-  ( ph  ->  ( ( B F A ) F ( C F D ) )  =  ( ( B F C ) F ( A F D ) ) )
94, 1, 2caovcomd 6143 . . 3  |-  ( ph  ->  ( B F A )  =  ( A F B ) )
109oveq1d 5996 . 2  |-  ( ph  ->  ( ( B F A ) F ( C F D ) )  =  ( ( A F B ) F ( C F D ) ) )
114, 1, 3caovcomd 6143 . . 3  |-  ( ph  ->  ( B F C )  =  ( C F B ) )
1211oveq1d 5996 . 2  |-  ( ph  ->  ( ( B F C ) F ( A F D ) )  =  ( ( C F B ) F ( A F D ) ) )
138, 10, 123eqtr3d 2406 1  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( ( C F B ) F ( A F D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715  (class class class)co 5981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-ov 5984
  Copyright terms: Public domain W3C validator