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Theorem ofldtos 32931
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
StepHypRef Expression
1 isofld 32922 . . . 4 (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
21simprbi 496 . . 3 (𝐹 ∈ oField → 𝐹 ∈ oRing)
3 orngogrp 32921 . . 3 (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4 isogrp 32723 . . . 4 (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
54simprbi 496 . . 3 (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
62, 3, 53syl 18 . 2 (𝐹 ∈ oField → 𝐹 ∈ oMnd)
7 omndtos 32726 . 2 (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
86, 7syl 17 1 (𝐹 ∈ oField → 𝐹 ∈ Toset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Tosetctos 18378  Grpcgrp 18860  Fieldcfield 20585  oMndcomnd 32718  oGrpcogrp 32719  oRingcorng 32915  oFieldcofld 32916
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 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-omnd 32720  df-ogrp 32721  df-orng 32917  df-ofld 32918
This theorem is referenced by:  ofldchr  32934
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