MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caov4d Structured version   Unicode version

Theorem caov4d 6263
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 6260 . . 3  |-  ( ph  ->  ( B F ( C F D ) )  =  ( C F ( B F D ) ) )
76oveq2d 6089 . 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 6232 . . 3  |-  ( ph  ->  ( C F D )  e.  S )
115, 8, 1, 10caovassd 6238 . 2  |-  ( ph  ->  ( ( A F B ) F ( C F D ) )  =  ( A F ( B F ( C F D ) ) ) )
129, 1, 3caovcld 6232 . . 3  |-  ( ph  ->  ( B F D )  e.  S )
135, 8, 2, 12caovassd 6238 . 2  |-  ( ph  ->  ( ( A F C ) F ( B F D ) )  =  ( A F ( C F ( B F D ) ) ) )
147, 11, 133eqtr4d 2477 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725  (class class class)co 6073
This theorem is referenced by:  caov411d  6264  caov42d  6265
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
  Copyright terms: Public domain W3C validator