MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfld Unicode version

Theorem isfld 15537
Description: A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Assertion
Ref Expression
isfld  |-  ( R  e. Field 
<->  ( R  e.  DivRing  /\  R  e.  CRing ) )

Proof of Theorem isfld
StepHypRef Expression
1 df-field 15531 . 2  |- Field  =  (
DivRing  i^i  CRing )
21elin2 3372 1  |-  ( R  e. Field 
<->  ( R  e.  DivRing  /\  R  e.  CRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1696   CRingccrg 15354   DivRingcdr 15528  Fieldcfield 15529
This theorem is referenced by:  fldpropd  15556  fldidom  16062  fiidomfld  16065  ply1pid  19581  lgseisenlem3  20606  lgseisenlem4  20607
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-field 15531
  Copyright terms: Public domain W3C validator