Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  flddivrng Structured version   Visualization version   GIF version

Theorem flddivrng 35146
 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 35139 . . 3 Fld = (DivRingOps ∩ Com2)
2 inss1 4208 . . 3 (DivRingOps ∩ Com2) ⊆ DivRingOps
31, 2eqsstri 4004 . 2 Fld ⊆ DivRingOps
43sseli 3966 1 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2106   ∩ cin 3938  DivRingOpscdrng 35095  Com2ccm2 35136  Fldcfld 35138 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-v 3501  df-in 3946  df-ss 3955  df-fld 35139 This theorem is referenced by:  isfld2  35152  isfldidl  35215
 Copyright terms: Public domain W3C validator