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Theorem isfld2 29994
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 25081 . . 3  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
2 fldcrng 29993 . . 3  |-  ( K  e.  Fld  ->  K  e. CRingOps )
31, 2jca 532 . 2  |-  ( K  e.  Fld  ->  ( K  e.  DivRingOps  /\  K  e. CRingOps )
)
4 iscrngo 29986 . . . 4  |-  ( K  e. CRingOps 
<->  ( K  e.  RingOps  /\  K  e.  Com2 ) )
54simprbi 464 . . 3  |-  ( K  e. CRingOps  ->  K  e.  Com2 )
6 elin 3682 . . . . 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 25080 . . . 4  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
97, 8syl6eleqr 2561 . . 3  |-  ( ( K  e.  DivRingOps  /\  K  e. 
Com2 )  ->  K  e.  Fld )
105, 9sylan2 474 . 2  |-  ( ( K  e.  DivRingOps  /\  K  e. CRingOps )  ->  K  e.  Fld )
113, 10impbii 188 1  |-  ( K  e.  Fld  <->  ( K  e. 
DivRingOps 
/\  K  e. CRingOps )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1762    i^i cin 3470   RingOpscrngo 25041   DivRingOpscdrng 25071   Com2ccm2 25076   Fldcfld 25079  CRingOpsccring 29984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-iota 5544  df-fun 5583  df-fv 5589  df-1st 6776  df-2nd 6777  df-drngo 25072  df-fld 25080  df-crngo 29985
This theorem is referenced by:  flddmn  30047  isfldidl  30057
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