Theorem isfld 21044

Description: A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)

Assertion

Ref	Expression
isfld	⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))

Proof of Theorem isfld

Step	Hyp	Ref	Expression
1		df-field 21034	. 2 ⊢ Field = (DivRing ∩ CRing)
2	1	elin2 4103	1 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  CRingccrg 20511  DivRingcdr 21031  Fieldcfield 21032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-field 21034

This theorem is used by:  flddrngd  21045  fldcrngd  21046  fldpropd  21079  fldidom  21080  fiidomfld  21083  rng1nfld  21087  fldcat  21091  fldsdrgfld  21106  primefld  21113  ofldlt1  21183  subofld  21185  isfieldidl  21591  ofldchr  21937  refld  21980  frlmphllem  22141  frlmphl  22142  matunitlindflem1  23049  matunitlindflem2  23050  matunitlindf  23051  recvs  25522  rrxcph  25768  rrx0  25773  ply1pid  26556  lgseisenlem3  27761  lgseisenlem4  27762  isarchiofld  33859  qfld  33963  fracfld  33974  fldgenfld  33986  cnfldfld  34007  reofld  34008  rearchi  34011  qsfld  34126  srafldlvec  34329  assafld  34380  ccfldextrr  34389  fldextsralvec  34398  extdgcl  34399  extdggt0  34400  fldextid  34402  extdgid  34403  extdgmul  34406  extdg1id  34409  ccfldsrarelvec  34414  2sqr3minply  34523  qqhrhm  34732  fldhmf1  43406  aks6d1c1p2  43425  aks6d1c2lem4  43443  aks6d1c5lem3  43453  aks6d1c5lem2  43454  aks6d1c6lem1  43486  ricfld  43860  fldcatALTV  49701  veroquadmodzerod  51412  veroquadnolindfd  51413  veroquaddetzerod  51414