Theorem isfld 18756

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 18750	. 2 |- Field = ( DivRing i^i CRing )
2	1	elin2 3801	1 |- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   CRingccrg 18548   DivRingcdr 18747  Fieldcfield 18748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-field 18750

This theorem is referenced by:  fldpropd  18775  rng1nfld  19278  fldidom  19305  fiidomfld  19308  refld  19965  recrng  19967  frlmphllem  20119  frlmphl  20120  rrxcph  23180  ply1pid  23939  lgseisenlem3  25102  lgseisenlem4  25103  ofldlt1  29813  ofldchr  29814  subofld  29816  isarchiofld  29817  reofld  29840  rearchi  29842  qqhrhm  30033  matunitlindflem1  33405  matunitlindflem2  33406  matunitlindf  33407  fldcat  42082  fldcatALTV  42100