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 33469
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 33462 . . 3 Fld = (DivRingOps ∩ Com2)
2 inss1 3817 . . 3 (DivRingOps ∩ Com2) ⊆ DivRingOps
31, 2eqsstri 3620 . 2 Fld ⊆ DivRingOps
43sseli 3584 1 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cin 3559  DivRingOpscdrng 33418  Com2ccm2 33459  Fldcfld 33461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-in 3567  df-ss 3574  df-fld 33462
This theorem is referenced by:  isfld2  33475  isfldidl  33538
  Copyright terms: Public domain W3C validator