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Theorem omndtos 30763
 Description: A left-ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)
Assertion
Ref Expression
omndtos (𝑀 ∈ oMnd → 𝑀 ∈ Toset)

Proof of Theorem omndtos
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2798 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2798 . . 3 (+g𝑀) = (+g𝑀)
3 eqid 2798 . . 3 (le‘𝑀) = (le‘𝑀)
41, 2, 3isomnd 30759 . 2 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)∀𝑐 ∈ (Base‘𝑀)(𝑎(le‘𝑀)𝑏 → (𝑎(+g𝑀)𝑐)(le‘𝑀)(𝑏(+g𝑀)𝑐))))
54simp2bi 1143 1 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  ∀wral 3106   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  +gcplusg 16559  lecple 16566  Tosetctos 17637  Mndcmnd 17905  oMndcomnd 30755 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-omnd 30757 This theorem is referenced by:  omndadd2d  30766  omndadd2rd  30767  submomnd  30768  omndmul2  30770  omndmul  30772  gsumle  30782  isarchi3  30873  archirng  30874  archirngz  30875  archiabllem1a  30877  archiabllem1b  30878  archiabllem2a  30880  archiabllem2c  30881  archiabllem2b  30882  archiabl  30884  orngsqr  30935  ofldtos  30942
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