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Theorem iscom2 21079
Description: A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
Assertion
Ref Expression
iscom2  |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  Com2  <->  A. a  e.  ran  G A. b  e.  ran  G ( a H b )  =  ( b H a ) ) )
Distinct variable groups:    G, a,
b    H, a, b
Allowed substitution hints:    A( a, b)    B( a, b)

Proof of Theorem iscom2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-com2 21078 . . . 4  |-  Com2  =  { <. x ,  y
>.  |  A. a  e.  ran  x A. b  e.  ran  x ( a y b )  =  ( b y a ) }
21a1i 10 . . 3  |-  ( ( G  e.  A  /\  H  e.  B )  ->  Com2  =  { <. x ,  y >.  |  A. a  e.  ran  x A. b  e.  ran  x ( a y b )  =  ( b y a ) } )
32eleq2d 2350 . 2  |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  Com2  <->  <. G ,  H >.  e.  { <. x ,  y >.  |  A. a  e.  ran  x A. b  e.  ran  x ( a y b )  =  ( b y a ) } ) )
4 rneq 4904 . . . 4  |-  ( x  =  G  ->  ran  x  =  ran  G )
54raleqdv 2742 . . . 4  |-  ( x  =  G  ->  ( A. b  e.  ran  x ( a y b )  =  ( b y a )  <->  A. b  e.  ran  G ( a y b )  =  ( b y a ) ) )
64, 5raleqbidv 2748 . . 3  |-  ( x  =  G  ->  ( A. a  e.  ran  x A. b  e.  ran  x ( a y b )  =  ( b y a )  <->  A. a  e.  ran  G A. b  e.  ran  G ( a y b )  =  ( b y a ) ) )
7 oveq 5864 . . . . 5  |-  ( y  =  H  ->  (
a y b )  =  ( a H b ) )
8 oveq 5864 . . . . 5  |-  ( y  =  H  ->  (
b y a )  =  ( b H a ) )
97, 8eqeq12d 2297 . . . 4  |-  ( y  =  H  ->  (
( a y b )  =  ( b y a )  <->  ( a H b )  =  ( b H a ) ) )
1092ralbidv 2585 . . 3  |-  ( y  =  H  ->  ( A. a  e.  ran  G A. b  e.  ran  G ( a y b )  =  ( b y a )  <->  A. a  e.  ran  G A. b  e.  ran  G ( a H b )  =  ( b H a ) ) )
116, 10opelopabg 4283 . 2  |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  { <. x ,  y >.  |  A. a  e.  ran  x A. b  e.  ran  x ( a y b )  =  ( b y a ) }  <->  A. a  e.  ran  G A. b  e.  ran  G ( a H b )  =  ( b H a ) ) )
123, 11bitrd 244 1  |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  Com2  <->  A. a  e.  ran  G A. b  e.  ran  G ( a H b )  =  ( b H a ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   {copab 4076   ran crn 4690  (class class class)co 5858   Com2ccm2 21077
This theorem is referenced by:  isfldOLD  25426  zintdom  25438  iscrngo2  26623  iscringd  26624
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  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-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263  df-ov 5861  df-com2 21078
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