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Theorem isfld2 26606
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 21995 . . 3  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
2 fldcrng 26605 . . 3  |-  ( K  e.  Fld  ->  K  e. CRingOps )
31, 2jca 519 . 2  |-  ( K  e.  Fld  ->  ( K  e.  DivRingOps  /\  K  e. CRingOps )
)
4 iscrngo 26598 . . . 4  |-  ( K  e. CRingOps 
<->  ( K  e.  RingOps  /\  K  e.  Com2 ) )
54simprbi 451 . . 3  |-  ( K  e. CRingOps  ->  K  e.  Com2 )
6 elin 3522 . . . . 5  |-  ( K  e.  ( DivRingOps  i^i  Com2 )  <->  ( K  e.  DivRingOps  /\  K  e. 
Com2 ) )
76biimpri 198 . . . 4  |-  ( ( K  e.  DivRingOps  /\  K  e. 
Com2 )  ->  K  e.  ( DivRingOps  i^i  Com2 ) )
8 df-fld 21994 . . . 4  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
97, 8syl6eleqr 2526 . . 3  |-  ( ( K  e.  DivRingOps  /\  K  e. 
Com2 )  ->  K  e.  Fld )
105, 9sylan2 461 . 2  |-  ( ( K  e.  DivRingOps  /\  K  e. CRingOps )  ->  K  e.  Fld )
113, 10impbii 181 1  |-  ( K  e.  Fld  <->  ( K  e. 
DivRingOps 
/\  K  e. CRingOps )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725    i^i cin 3311   RingOpscrngo 21955   DivRingOpscdrng 21985   Com2ccm2 21990   Fldcfld 21993  CRingOpsccring 26596
This theorem is referenced by:  flddmn  26659  isfldidl  26669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-1st 6341  df-2nd 6342  df-drngo 21986  df-fld 21994  df-crngo 26597
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