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Theorem fldcrng 26615
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 2437 . . . . 5  |-  ( 1st `  K )  =  ( 1st `  K )
2 eqid 2437 . . . . 5  |-  ( 2nd `  K )  =  ( 2nd `  K )
3 eqid 2437 . . . . 5  |-  ran  ( 1st `  K )  =  ran  ( 1st `  K
)
4 eqid 2437 . . . . 5  |-  (GId `  ( 1st `  K ) )  =  (GId `  ( 1st `  K ) )
51, 2, 3, 4drngoi 21996 . . . 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 447 . . 3  |-  ( K  e.  DivRingOps  ->  K  e.  RingOps )
76anim1i 553 . 2  |-  ( ( K  e.  DivRingOps  /\  K  e. 
Com2 )  ->  ( K  e.  RingOps  /\  K  e.  Com2 ) )
8 df-fld 22003 . . 3  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
98elin2 3532 . 2  |-  ( K  e.  Fld  <->  ( K  e. 
DivRingOps 
/\  K  e.  Com2 ) )
10 iscrngo 26608 . 2  |-  ( K  e. CRingOps 
<->  ( K  e.  RingOps  /\  K  e.  Com2 ) )
117, 9, 103imtr4i 259 1  |-  ( K  e.  Fld  ->  K  e. CRingOps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726    \ cdif 3318   {csn 3815    X. cxp 4877   ran crn 4880    |` cres 4881   ` cfv 5455   1stc1st 6348   2ndc2nd 6349   GrpOpcgr 21775  GIdcgi 21776   RingOpscrngo 21964   DivRingOpscdrng 21994   Com2ccm2 21999   Fldcfld 22002  CRingOpsccring 26606
This theorem is referenced by:  isfld2  26616  isfldidl  26679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-iota 5419  df-fun 5457  df-fv 5463  df-1st 6350  df-2nd 6351  df-drngo 21995  df-fld 22003  df-crngo 26607
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