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Theorem iscom2 21095
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 21094 . . . 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 2363 . 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 4920 . . . 4  |-  ( x  =  G  ->  ran  x  =  ran  G )
54raleqdv 2755 . . . 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 2761 . . 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 5880 . . . . 5  |-  ( y  =  H  ->  (
a y b )  =  ( a H b ) )
8 oveq 5880 . . . . 5  |-  ( y  =  H  ->  (
b y a )  =  ( b H a ) )
97, 8eqeq12d 2310 . . . 4  |-  ( y  =  H  ->  (
( a y b )  =  ( b y a )  <->  ( a H b )  =  ( b H a ) ) )
1092ralbidv 2598 . . 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 4299 . 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 1632    e. wcel 1696   A.wral 2556   <.cop 3656   {copab 4092   ran crn 4706  (class class class)co 5874   Com2ccm2 21093
This theorem is referenced by:  isfldOLD  25529  zintdom  25541  iscrngo2  26726  iscringd  26727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-iota 5235  df-fv 5279  df-ov 5877  df-com2 21094
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