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Theorem omndtos 32201
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 2733 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2733 . . 3 (+g𝑀) = (+g𝑀)
3 eqid 2733 . . 3 (le‘𝑀) = (le‘𝑀)
41, 2, 3isomnd 32197 . 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 1147 1 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wral 3062   class class class wbr 5147  cfv 6540  (class class class)co 7404  Basecbs 17140  +gcplusg 17193  lecple 17200  Tosetctos 18365  Mndcmnd 18621  oMndcomnd 32193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7407  df-omnd 32195
This theorem is referenced by:  omndadd2d  32204  omndadd2rd  32205  submomnd  32206  omndmul2  32208  omndmul  32210  gsumle  32220  isarchi3  32311  archirng  32312  archirngz  32313  archiabllem1a  32315  archiabllem1b  32316  archiabllem2a  32318  archiabllem2c  32319  archiabllem2b  32320  archiabl  32322  orngsqr  32391  ofldtos  32398
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