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Theorem isfld2 31684
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 25831 . . 3  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
2 fldcrng 31683 . . 3  |-  ( K  e.  Fld  ->  K  e. CRingOps )
31, 2jca 530 . 2  |-  ( K  e.  Fld  ->  ( K  e.  DivRingOps  /\  K  e. CRingOps )
)
4 iscrngo 31676 . . . 4  |-  ( K  e. CRingOps 
<->  ( K  e.  RingOps  /\  K  e.  Com2 ) )
54simprbi 462 . . 3  |-  ( K  e. CRingOps  ->  K  e.  Com2 )
6 elin 3626 . . . . 5  |-  ( K  e.  ( DivRingOps  i^i  Com2 )  <->  ( K  e.  DivRingOps  /\  K  e. 
Com2 ) )
76biimpri 206 . . . 4  |-  ( ( K  e.  DivRingOps  /\  K  e. 
Com2 )  ->  K  e.  ( DivRingOps  i^i  Com2 ) )
8 df-fld 25830 . . . 4  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
97, 8syl6eleqr 2501 . . 3  |-  ( ( K  e.  DivRingOps  /\  K  e. 
Com2 )  ->  K  e.  Fld )
105, 9sylan2 472 . 2  |-  ( ( K  e.  DivRingOps  /\  K  e. CRingOps )  ->  K  e.  Fld )
113, 10impbii 187 1  |-  ( K  e.  Fld  <->  ( K  e. 
DivRingOps 
/\  K  e. CRingOps )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    e. wcel 1842    i^i cin 3413   RingOpscrngo 25791   DivRingOpscdrng 25821   Com2ccm2 25826   Fldcfld 25829  CRingOpsccring 31674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-iota 5533  df-fun 5571  df-fv 5577  df-1st 6784  df-2nd 6785  df-drngo 25822  df-fld 25830  df-crngo 31675
This theorem is referenced by:  flddmn  31737  isfldidl  31747
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