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Theorem isfld2 26630
Description: The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
isfld2  |-  ( K  e.  Fld  <->  ( K  e. 
DivRingOps 
/\  K  e. CRingOps )
)

Proof of Theorem isfld2
StepHypRef Expression
1 flddivrng 21082 . . 3  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
2 fldcrng 26629 . . 3  |-  ( K  e.  Fld  ->  K  e. CRingOps )
31, 2jca 518 . 2  |-  ( K  e.  Fld  ->  ( K  e.  DivRingOps  /\  K  e. CRingOps )
)
4 iscrngo 26622 . . . 4  |-  ( K  e. CRingOps 
<->  ( K  e.  RingOps  /\  K  e.  Com2 ) )
54simprbi 450 . . 3  |-  ( K  e. CRingOps  ->  K  e.  Com2 )
6 elin 3358 . . . . 5  |-  ( K  e.  ( DivRingOps  i^i  Com2 )  <->  ( K  e.  DivRingOps  /\  K  e. 
Com2 ) )
76biimpri 197 . . . 4  |-  ( ( K  e.  DivRingOps  /\  K  e. 
Com2 )  ->  K  e.  ( DivRingOps  i^i  Com2 ) )
8 df-fld 21081 . . . 4  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
97, 8syl6eleqr 2374 . . 3  |-  ( ( K  e.  DivRingOps  /\  K  e. 
Com2 )  ->  K  e.  Fld )
105, 9sylan2 460 . 2  |-  ( ( K  e.  DivRingOps  /\  K  e. CRingOps )  ->  K  e.  Fld )
113, 10impbii 180 1  |-  ( K  e.  Fld  <->  ( K  e. 
DivRingOps 
/\  K  e. CRingOps )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684    i^i cin 3151   RingOpscrngo 21042   DivRingOpscdrng 21072   Com2ccm2 21077   Fldcfld 21080  CRingOpsccring 26620
This theorem is referenced by:  flddmn  26683  isfldidl  26693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-1st 6122  df-2nd 6123  df-drngo 21073  df-fld 21081  df-crngo 26621
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