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Theorem fldcrng 26732
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 2296 . . . . 5  |-  ( 1st `  K )  =  ( 1st `  K )
2 eqid 2296 . . . . 5  |-  ( 2nd `  K )  =  ( 2nd `  K )
3 eqid 2296 . . . . 5  |-  ran  ( 1st `  K )  =  ran  ( 1st `  K
)
4 eqid 2296 . . . . 5  |-  (GId `  ( 1st `  K ) )  =  (GId `  ( 1st `  K ) )
51, 2, 3, 4drngoi 21090 . . . 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 21097 . . 3  |-  Fld  =  (
DivRingOps 
i^i  Com2 )
98elin2 3372 . 2  |-  ( K  e.  Fld  <->  ( K  e. 
DivRingOps 
/\  K  e.  Com2 ) )
10 iscrngo 26725 . 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 1696    \ cdif 3162   {csn 3653    X. cxp 4703   ran crn 4706    |` cres 4707   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   GrpOpcgr 20869  GIdcgi 20870   RingOpscrngo 21058   DivRingOpscdrng 21088   Com2ccm2 21093   Fldcfld 21096  CRingOpsccring 26723
This theorem is referenced by:  isfld2  26733  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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