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Theorem flddivrng 32968
 Description: A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
flddivrng (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

Proof of Theorem flddivrng
StepHypRef Expression
1 df-fld 32961 . . 3 Fld = (DivRingOps ∩ Com2)
2 inss1 3795 . . 3 (DivRingOps ∩ Com2) ⊆ DivRingOps
31, 2eqsstri 3598 . 2 Fld ⊆ DivRingOps
43sseli 3564 1 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ∩ cin 3539  DivRingOpscdrng 32917  Com2ccm2 32958  Fldcfld 32960 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-fld 32961 This theorem is referenced by:  isfld2  32974  isfldidl  33037
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