Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofldfld Structured version   Visualization version   GIF version

Theorem ofldfld 31099
Description: An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Assertion
Ref Expression
ofldfld (𝐹 ∈ oField → 𝐹 ∈ Field)

Proof of Theorem ofldfld
StepHypRef Expression
1 isofld 31091 . 2 (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
21simplbi 501 1 (𝐹 ∈ oField → 𝐹 ∈ Field)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  Fieldcfield 19635  oRingcorng 31084  oFieldcofld 31085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3402  df-in 3860  df-ofld 31087
This theorem is referenced by:  ofldlt1  31102  ofldchr  31103
  Copyright terms: Public domain W3C validator