Theorem ofldtos 31510

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 31501	. . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simprbi 497	. . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing)
3		orngogrp 31500	. . 3 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4		isogrp 31328	. . . 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 31331	. 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
8	6, 7	syl 17	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2106  Tosetctos 18134  Grpcgrp 18577  Fieldcfield 19992  oMndcomnd 31323  oGrpcogrp 31324  oRingcorng 31494  oFieldcofld 31495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-omnd 31325  df-ogrp 31326  df-orng 31496  df-ofld 31497

This theorem is referenced by:  ofldchr  31513