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Theorem isfldOLD 25529
Description: The predicate "is a field". (Contributed by FL, 6-Sep-2009.)
Assertion
Ref Expression
isfldOLD  |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  Fld  <->  ( <. G ,  H >.  e.  DivRingOps  /\  A. x  e.  ran  G A. y  e.  ran  G ( x H y )  =  ( y H x ) ) ) )
Distinct variable groups:    x, G, y    x, H, y
Allowed substitution hints:    A( x, y)    B( x, y)

Proof of Theorem isfldOLD
StepHypRef Expression
1 df-fld 21097 . . . 4  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
21a1i 10 . . 3  |-  ( ( G  e.  A  /\  H  e.  B )  ->  Fld  =  ( DivRingOps  i^i  Com2 ) )
32eleq2d 2363 . 2  |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  Fld  <->  <. G ,  H >.  e.  ( DivRingOps  i^i  Com2 ) ) )
4 elin 3371 . . 3  |-  ( <. G ,  H >.  e.  ( DivRingOps  i^i  Com2 )  <->  ( <. G ,  H >.  e.  DivRingOps  /\  <. G ,  H >.  e.  Com2 )
)
54a1i 10 . 2  |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  ( DivRingOps  i^i  Com2 )  <->  (
<. G ,  H >.  e.  DivRingOps  /\  <. G ,  H >.  e.  Com2 ) ) )
6 iscom2 21095 . . 3  |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  Com2  <->  A. x  e.  ran  G A. y  e.  ran  G ( x H y )  =  ( y H x ) ) )
76anbi2d 684 . 2  |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( ( <. G ,  H >.  e.  DivRingOps  /\  <. G ,  H >.  e.  Com2 )  <->  (
<. G ,  H >.  e.  DivRingOps  /\  A. x  e.  ran  G A. y  e.  ran  G ( x H y )  =  ( y H x ) ) ) )
83, 5, 73bitrd 270 1  |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  Fld  <->  ( <. G ,  H >.  e.  DivRingOps  /\  A. x  e.  ran  G A. y  e.  ran  G ( x H y )  =  ( y H x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164   <.cop 3656   ran crn 4706  (class class class)co 5874   DivRingOpscdrng 21088   Com2ccm2 21093   Fldcfld 21096
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  df-fld 21097
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