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Theorem flddivrng 36143
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 36136 . . 3 Fld = (DivRingOps ∩ Com2)
2 inss1 4163 . . 3 (DivRingOps ∩ Com2) ⊆ DivRingOps
31, 2eqsstri 3955 . 2 Fld ⊆ DivRingOps
43sseli 3917 1 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cin 3886  DivRingOpscdrng 36092  Com2ccm2 36133  Fldcfld 36135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3432  df-in 3894  df-ss 3904  df-fld 36136
This theorem is referenced by:  isfld2  36149  isfldidl  36212
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