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Theorem isfld2 26733
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 21098 . . 3  |-  ( K  e.  Fld  ->  K  e. 
DivRingOps )
2 fldcrng 26732 . . 3  |-  ( K  e.  Fld  ->  K  e. CRingOps )
31, 2jca 518 . 2  |-  ( K  e.  Fld  ->  ( K  e.  DivRingOps  /\  K  e. CRingOps )
)
4 iscrngo 26725 . . . 4  |-  ( K  e. CRingOps 
<->  ( K  e.  RingOps  /\  K  e.  Com2 ) )
54simprbi 450 . . 3  |-  ( K  e. CRingOps  ->  K  e.  Com2 )
6 elin 3371 . . . . 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 21097 . . . 4  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
97, 8syl6eleqr 2387 . . 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 1696    i^i cin 3164   RingOpscrngo 21058   DivRingOpscdrng 21088   Com2ccm2 21093   Fldcfld 21096  CRingOpsccring 26723
This theorem is referenced by:  flddmn  26786  isfldidl  26796
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-sbc 3005  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-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138  df-2nd 6139  df-drngo 21089  df-fld 21097  df-crngo 26724
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