Theorem ofldtos 20789

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 20780	. . 3 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simprbi 496	. 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing)
3		orngogrp 20779	. 2 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4		isogrp 20037	. . 3 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
5	4	simprbi 496	. 2 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
6		omndtos 20040	. 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
7	2, 3, 5, 6	4syl 19	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2111  Tosetctos 18320  Grpcgrp 18846  oMndcomnd 20032  oGrpcogrp 20033  Fieldcfield 20646  oRingcorng 20773  oFieldcofld 20774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-ov 7349  df-omnd 20034  df-ogrp 20035  df-orng 20775  df-ofld 20776

This theorem is referenced by:  ofldchr  21514