Theorem ofldtos 33050

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 33041	. . . 4 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simprbi 495	. . 3 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing)
3		orngogrp 33040	. . 3 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4		isogrp 32803	. . . 4 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
5	4	simprbi 495	. . 3 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
6	2, 3, 5	3syl 18	. 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oMnd)
7		omndtos 32806	. 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
8	6, 7	syl 17	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2098  Tosetctos 18415  Grpcgrp 18897  Fieldcfield 20632  oMndcomnd 32798  oGrpcogrp 32799  oRingcorng 33034  oFieldcofld 33035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-omnd 32800  df-ogrp 32801  df-orng 33036  df-ofld 33037

This theorem is referenced by:  ofldchr  33053