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Theorem fldcrng 26629
Description: A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
Assertion
Ref Expression
fldcrng  |-  ( K  e.  Fld  ->  K  e. CRingOps )

Proof of Theorem fldcrng
StepHypRef Expression
1 eqid 2283 . . . . 5  |-  ( 1st `  K )  =  ( 1st `  K )
2 eqid 2283 . . . . 5  |-  ( 2nd `  K )  =  ( 2nd `  K )
3 eqid 2283 . . . . 5  |-  ran  ( 1st `  K )  =  ran  ( 1st `  K
)
4 eqid 2283 . . . . 5  |-  (GId `  ( 1st `  K ) )  =  (GId `  ( 1st `  K ) )
51, 2, 3, 4drngoi 21074 . . . 4  |-  ( K  e.  DivRingOps  ->  ( K  e.  RingOps 
/\  ( ( 2nd `  K )  |`  (
( ran  ( 1st `  K )  \  {
(GId `  ( 1st `  K ) ) } )  X.  ( ran  ( 1st `  K
)  \  { (GId `  ( 1st `  K
) ) } ) ) )  e.  GrpOp ) )
65simpld 445 . . 3  |-  ( K  e.  DivRingOps  ->  K  e.  RingOps )
76anim1i 551 . 2  |-  ( ( K  e.  DivRingOps  /\  K  e. 
Com2 )  ->  ( K  e.  RingOps  /\  K  e.  Com2 ) )
8 df-fld 21081 . . 3  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
98elin2 3359 . 2  |-  ( K  e.  Fld  <->  ( K  e. 
DivRingOps 
/\  K  e.  Com2 ) )
10 iscrngo 26622 . 2  |-  ( K  e. CRingOps 
<->  ( K  e.  RingOps  /\  K  e.  Com2 ) )
117, 9, 103imtr4i 257 1  |-  ( K  e.  Fld  ->  K  e. CRingOps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    \ cdif 3149   {csn 3640    X. cxp 4687   ran crn 4690    |` cres 4691   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854   RingOpscrngo 21042   DivRingOpscdrng 21072   Com2ccm2 21077   Fldcfld 21080  CRingOpsccring 26620
This theorem is referenced by:  isfld2  26630  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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