Theorem ofldtos 29142

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 29133	. . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simprbi 479	. . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing)
3		orngogrp 29132	. . 3 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4		isogrp 29033	. . . 4 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
5	4	simprbi 479	. . 3 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
6	2, 3, 5	3syl 18	. 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oMnd)
7		omndtos 29036	. 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
8	6, 7	syl 17	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 1977  Tosetctos 16856  Grpcgrp 17245  Fieldcfield 18571  oMndcomnd 29028  oGrpcogrp 29029  oRingcorng 29126  oFieldcofld 29127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-omnd 29030  df-ogrp 29031  df-orng 29128  df-ofld 29129

This theorem is referenced by:  ofldchr  29145