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Theorem isfld 19440
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
StepHypRef Expression
1 df-field 19434 . 2 Field = (DivRing ∩ CRing)
21elin2 4171 1 (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2105  CRingccrg 19227  DivRingcdr 19431  Fieldcfield 19432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-field 19434
This theorem is referenced by:  fldpropd  19459  primefld  19513  rng1nfld  19979  fldidom  20006  fiidomfld  20009  refld  20691  recrng  20693  frlmphllem  20852  frlmphl  20853  rrxcph  23922  rrx0  23927  ply1pid  24700  lgseisenlem3  25880  lgseisenlem4  25881  ofldlt1  30813  ofldchr  30814  subofld  30816  isarchiofld  30817  reofld  30840  rearchi  30842  srafldlvec  30890  rgmoddim  30907  ccfldextrr  30937  fldextsralvec  30944  extdgcl  30945  extdggt0  30946  fldextid  30948  extdgmul  30950  extdg1id  30952  ccfldsrarelvec  30955  qqhrhm  31129  matunitlindflem1  34769  matunitlindflem2  34770  matunitlindf  34771  fldcat  44281  fldcatALTV  44299
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