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Theorem omndtos 29833
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 2651 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2651 . . 3 (+g𝑀) = (+g𝑀)
3 eqid 2651 . . 3 (le‘𝑀) = (le‘𝑀)
41, 2, 3isomnd 29829 . 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 1097 1 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wral 2941   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  lecple 15995  Tosetctos 17080  Mndcmnd 17341  oMndcomnd 29825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-omnd 29827
This theorem is referenced by:  omndadd2d  29836  omndadd2rd  29837  submomnd  29838  omndmul2  29840  omndmul  29842  isarchi3  29869  archirng  29870  archirngz  29871  archiabllem1a  29873  archiabllem1b  29874  archiabllem2a  29876  archiabllem2c  29877  archiabllem2b  29878  archiabl  29880  gsumle  29907  orngsqr  29932  ofldtos  29939
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