Theorem ofldtos 32417

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 32408	. . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simprbi 497	. . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing)
3		orngogrp 32407	. . 3 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4		isogrp 32207	. . . 4 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
5	4	simprbi 497	. . 3 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
6	2, 3, 5	3syl 18	. 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oMnd)
7		omndtos 32210	. 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
8	6, 7	syl 17	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2106  Tosetctos 18365  Grpcgrp 18815  Fieldcfield 20308  oMndcomnd 32202  oGrpcogrp 32203  oRingcorng 32401  oFieldcofld 32402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-omnd 32204  df-ogrp 32205  df-orng 32403  df-ofld 32404

This theorem is referenced by:  ofldchr  32420