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Theorem iscom2 25612
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 25611 . . . 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 2524 . 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 5217 . . . 4  |-  ( x  =  G  ->  ran  x  =  ran  G )
54raleqdv 3057 . . . 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 3065 . . 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 6276 . . . . 5  |-  ( y  =  H  ->  (
a y b )  =  ( a H b ) )
8 oveq 6276 . . . . 5  |-  ( y  =  H  ->  (
b y a )  =  ( b H a ) )
97, 8eqeq12d 2476 . . . 4  |-  ( y  =  H  ->  (
( a y b )  =  ( b y a )  <->  ( a H b )  =  ( b H a ) ) )
1092ralbidv 2898 . . 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 4754 . 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 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   <.cop 4022   {copab 4496   ran crn 4989  (class class class)co 6270   Com2ccm2 25610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-cnv 4996  df-dm 4998  df-rn 4999  df-iota 5534  df-fv 5578  df-ov 6273  df-com2 25611
This theorem is referenced by:  iscrngo2  30635  iscringd  30636
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