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Theorem iscom 25333
Description: The predicate "is a commutative operation". (Contributed by FL, 5-Sep-2010.)
Hypothesis
Ref Expression
iscom.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
iscom  |-  ( G  e.  A  ->  ( G  e.  Com1  <->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
Distinct variable group:    x, G, y
Allowed substitution hints:    A( x, y)    X( x, y)

Proof of Theorem iscom
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 iscom.1 . . . . . 6  |-  X  =  dom  dom  G
2 dmeq 4879 . . . . . . . 8  |-  ( a  =  G  ->  dom  a  =  dom  G )
32eqcomd 2288 . . . . . . 7  |-  ( a  =  G  ->  dom  G  =  dom  a )
43dmeqd 4881 . . . . . 6  |-  ( a  =  G  ->  dom  dom 
G  =  dom  dom  a )
51, 4syl5req 2328 . . . . 5  |-  ( a  =  G  ->  dom  dom  a  =  X )
65eleq2d 2350 . . . 4  |-  ( a  =  G  ->  (
x  e.  dom  dom  a 
<->  x  e.  X ) )
75eleq2d 2350 . . . . . 6  |-  ( a  =  G  ->  (
y  e.  dom  dom  a 
<->  y  e.  X ) )
8 oveq 5864 . . . . . . 7  |-  ( a  =  G  ->  (
x a y )  =  ( x G y ) )
9 oveq 5864 . . . . . . 7  |-  ( a  =  G  ->  (
y a x )  =  ( y G x ) )
108, 9eqeq12d 2297 . . . . . 6  |-  ( a  =  G  ->  (
( x a y )  =  ( y a x )  <->  ( x G y )  =  ( y G x ) ) )
117, 10imbi12d 311 . . . . 5  |-  ( a  =  G  ->  (
( y  e.  dom  dom  a  ->  ( x
a y )  =  ( y a x ) )  <->  ( y  e.  X  ->  ( x G y )  =  ( y G x ) ) ) )
1211ralbidv2 2565 . . . 4  |-  ( a  =  G  ->  ( A. y  e.  dom  dom  a ( x a y )  =  ( y a x )  <->  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
136, 12imbi12d 311 . . 3  |-  ( a  =  G  ->  (
( x  e.  dom  dom  a  ->  A. y  e.  dom  dom  a (
x a y )  =  ( y a x ) )  <->  ( x  e.  X  ->  A. y  e.  X  ( x G y )  =  ( y G x ) ) ) )
1413ralbidv2 2565 . 2  |-  ( a  =  G  ->  ( A. x  e.  dom  dom  a A. y  e. 
dom  dom  a ( x a y )  =  ( y a x )  <->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
15 df-com1 25332 . 2  |-  Com1  =  { a  |  A. x  e.  dom  dom  a A. y  e.  dom  dom  a ( x a y )  =  ( y a x ) }
1614, 15elab2g 2916 1  |-  ( G  e.  A  ->  ( G  e.  Com1  <->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   dom cdm 4689  (class class class)co 5858   Com1ccm1 25331
This theorem is referenced by:  iscomb  25334  ablocomgrp  25342
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
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