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Theorem iscom2 24052
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 24051 . . . 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 5174 . . . 4  |-  ( x  =  G  ->  ran  x  =  ran  G )
54raleqdv 3029 . . . 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 3037 . . 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 6207 . . . . 5  |-  ( y  =  H  ->  (
a y b )  =  ( a H b ) )
8 oveq 6207 . . . . 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 2879 . . 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 4716 . 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 369    = wceq 1370    e. wcel 1758   A.wral 2799   <.cop 3992   {copab 4458   ran crn 4950  (class class class)co 6201   Com2ccm2 24050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-cnv 4957  df-dm 4959  df-rn 4960  df-iota 5490  df-fv 5535  df-ov 6204  df-com2 24051
This theorem is referenced by:  iscrngo2  28947  iscringd  28948
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