Theorem isfld 17200

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

Step	Hyp	Ref	Expression
1		df-field 17194	. 2 |- Field = ( DivRing i^i CRing )
2	1	elin2 3689	1 |- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) )

Syntax hints:   <-> wb 184   /\ wa 369   e. wcel 1767   CRingccrg 16996   DivRingcdr 17191  Fieldcfield 17192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-field 17194

This theorem is referenced by:  fldpropd  17219  rng1nfld  17713  fldidom  17741  fiidomfld  17744  refld  18438  recrng  18440  frlmphllem  18594  frlmphl  18595  rrxcph  21575  ply1pid  22331  lgseisenlem3  23370  lgseisenlem4  23371  ofldlt1  27482  ofldchr  27483  subofld  27485  isarchiofld  27486  reofld  27509  rearchi  27511  qqhrhm  27622