Theorem ofldtos 20945

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 20936	. . 3 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simprbi 502	. 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing)
3		orngogrp 20935	. 2 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4		isogrp 20185	. . 3 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
5	4	simprbi 502	. 2 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
6		omndtos 20188	. 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
7	2, 3, 5, 6	4syl 20	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2145  Tosetctos 18460  Grpcgrp 18990  oMndcomnd 20180  oGrpcogrp 20181  Fieldcfield 20805  oRingcorng 20929  oFieldcofld 20930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403  df-omnd 20182  df-ogrp 20183  df-orng 20931  df-ofld 20932

This theorem is referenced by:  ofldchr  21686