Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iscomb Unicode version

Theorem iscomb 25334
Description: The predicate "is a commutative operation". (Contributed by FL, 14-Sep-2010.)
Hypothesis
Ref Expression
iscom.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
iscomb  |-  ( ( G  e.  Com1  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )

Proof of Theorem iscomb
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscom.1 . . . . 5  |-  X  =  dom  dom  G
21iscom 25333 . . . 4  |-  ( G  e.  Com1  ->  ( G  e.  Com1  <->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
3 oveq1 5865 . . . . . . . 8  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
4 oveq2 5866 . . . . . . . 8  |-  ( x  =  A  ->  (
y G x )  =  ( y G A ) )
53, 4eqeq12d 2297 . . . . . . 7  |-  ( x  =  A  ->  (
( x G y )  =  ( y G x )  <->  ( A G y )  =  ( y G A ) ) )
6 oveq2 5866 . . . . . . . 8  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
7 oveq1 5865 . . . . . . . 8  |-  ( y  =  B  ->  (
y G A )  =  ( B G A ) )
86, 7eqeq12d 2297 . . . . . . 7  |-  ( y  =  B  ->  (
( A G y )  =  ( y G A )  <->  ( A G B )  =  ( B G A ) ) )
95, 8rspc2v 2890 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x )  ->  ( A G B )  =  ( B G A ) ) )
109ex 423 . . . . 5  |-  ( A  e.  X  ->  ( B  e.  X  ->  ( A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x )  -> 
( A G B )  =  ( B G A ) ) ) )
1110com3r 73 . . . 4  |-  ( A. x  e.  X  A. y  e.  X  (
x G y )  =  ( y G x )  ->  ( A  e.  X  ->  ( B  e.  X  -> 
( A G B )  =  ( B G A ) ) ) )
122, 11syl6bi 219 . . 3  |-  ( G  e.  Com1  ->  ( G  e.  Com1  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  =  ( B G A ) ) ) ) )
1312pm2.43i 43 . 2  |-  ( G  e.  Com1  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  =  ( B G A ) ) ) )
14133imp 1145 1  |-  ( ( G  e.  Com1  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   dom cdm 4689  (class class class)co 5858   Com1ccm1 25331
This theorem is referenced by:  reacomsmgrp1  25343  reacomsmgrp2  25344  reacomsmgrp3  25345  fprodadd  25352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861  df-com1 25332
  Copyright terms: Public domain W3C validator