Theorem ofldtos 21163

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 21154	. . 3 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simprbi 503	. 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing)
3		orngogrp 21153	. 2 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4		isogrp 20371	. . 3 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
5	4	simprbi 503	. 2 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
6		omndtos 20374	. 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
7	2, 3, 5, 6	4syl 20	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Tosetctos 18620  Grpcgrp 19176  oMndcomnd 20366  oGrpcogrp 20367  Fieldcfield 21014  oRingcorng 21147  oFieldcofld 21148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5264

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-ov 7449  df-omnd 20368  df-ogrp 20369  df-orng 21149  df-ofld 21150

This theorem is used by:  ofldchr  21915