Theorem ofldtos 30879

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 30870	. . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simprbi 499	. . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing)
3		orngogrp 30869	. . 3 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4		isogrp 30698	. . . 4 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
5	4	simprbi 499	. . 3 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
6	2, 3, 5	3syl 18	. 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oMnd)
7		omndtos 30701	. 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
8	6, 7	syl 17	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2110  Tosetctos 17637  Grpcgrp 18097  Fieldcfield 19497  oMndcomnd 30693  oGrpcogrp 30694  oRingcorng 30863  oFieldcofld 30864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7153  df-omnd 30695  df-ogrp 30696  df-orng 30865  df-ofld 30866

This theorem is referenced by:  ofldchr  30882