Theorem ofldtos 21136

Description: An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.)

Assertion

Ref	Expression
ofldtos	⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

Proof of Theorem ofldtos

Step	Hyp	Ref	Expression
1		isofld 21127	. . 3 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simprbi 503	. 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing)
3		orngogrp 21126	. 2 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4		isogrp 20344	. . 3 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
5	4	simprbi 503	. 2 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
6		omndtos 20347	. 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
7	2, 3, 5, 6	4syl 20	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Tosetctos 18594  Grpcgrp 19150  oMndcomnd 20339  oGrpcogrp 20340  Fieldcfield 20987  oRingcorng 21120  oFieldcofld 21121

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6547  df-ov 7424  df-omnd 20341  df-ogrp 20342  df-orng 21122  df-ofld 21123

This theorem is used by:  ofldchr  21888