MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flddrngd Structured version   Visualization version   GIF version

Theorem flddrngd 20640
Description: A field is a division ring. (Contributed by SN, 17-Jan-2025.)
Hypothesis
Ref Expression
flddrngd.1 (𝜑𝑅 ∈ Field)
Assertion
Ref Expression
flddrngd (𝜑𝑅 ∈ DivRing)

Proof of Theorem flddrngd
StepHypRef Expression
1 flddrngd.1 . 2 (𝜑𝑅 ∈ Field)
2 isfld 20639 . . 3 (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
32simplbi 496 . 2 (𝑅 ∈ Field → 𝑅 ∈ DivRing)
41, 3syl 17 1 (𝜑𝑅 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  CRingccrg 20178  DivRingcdr 20628  Fieldcfield 20629
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-in 3946  df-field 20631
This theorem is referenced by:  ply1asclunit  33316  ply1unit  33317  m1pmeq  33318  evls1fldgencl  33415  minplyirred  33438  algextdeglem2  33443  algextdeglem3  33444  algextdeglem4  33445  algextdeglem5  33446  algextdeglem7  33448  algextdeglem8  33449  aks6d1c5lem3  41664  aks6d1c5lem2  41665  prjcrv0  42122
  Copyright terms: Public domain W3C validator