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

Theorem flddivrng 38006
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 37999 . . 3 Fld = (DivRingOps ∩ Com2)
2 inss1 4237 . . 3 (DivRingOps ∩ Com2) ⊆ DivRingOps
31, 2eqsstri 4030 . 2 Fld ⊆ DivRingOps
43sseli 3979 1 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  cin 3950  DivRingOpscdrng 37955  Com2ccm2 37996  Fldcfld 37998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-ss 3968  df-fld 37999
This theorem is referenced by:  isfld2  38012  isfldidl  38075
  Copyright terms: Public domain W3C validator