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Theorem iscom2 22000
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 21999 . . . 4  |-  Com2  =  { <. x ,  y
>.  |  A. a  e.  ran  x A. b  e.  ran  x ( a y b )  =  ( b y a ) }
21a1i 11 . . 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 2503 . 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 5095 . . . 4  |-  ( x  =  G  ->  ran  x  =  ran  G )
54raleqdv 2910 . . . 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 2916 . . 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 6087 . . . . 5  |-  ( y  =  H  ->  (
a y b )  =  ( a H b ) )
8 oveq 6087 . . . . 5  |-  ( y  =  H  ->  (
b y a )  =  ( b H a ) )
97, 8eqeq12d 2450 . . . 4  |-  ( y  =  H  ->  (
( a y b )  =  ( b y a )  <->  ( a H b )  =  ( b H a ) ) )
1092ralbidv 2747 . . 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 4473 . 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 245 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   <.cop 3817   {copab 4265   ran crn 4879  (class class class)co 6081   Com2ccm2 21998
This theorem is referenced by:  iscrngo2  26608  iscringd  26609
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-cnv 4886  df-dm 4888  df-rn 4889  df-iota 5418  df-fv 5462  df-ov 6084  df-com2 21999
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