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Theorem ofldtos 29785
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 29776 . . . 4 (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
21simprbi 480 . . 3 (𝐹 ∈ oField → 𝐹 ∈ oRing)
3 orngogrp 29775 . . 3 (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4 isogrp 29676 . . . 4 (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
54simprbi 480 . . 3 (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
62, 3, 53syl 18 . 2 (𝐹 ∈ oField → 𝐹 ∈ oMnd)
7 omndtos 29679 . 2 (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
86, 7syl 17 1 (𝐹 ∈ oField → 𝐹 ∈ Toset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988  Tosetctos 17014  Grpcgrp 17403  Fieldcfield 18729  oMndcomnd 29671  oGrpcogrp 29672  oRingcorng 29769  oFieldcofld 29770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-omnd 29673  df-ogrp 29674  df-orng 29771  df-ofld 29772
This theorem is referenced by:  ofldchr  29788
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