Theorem omndtos 20348

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.

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2760	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2760	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 20344	. 2 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)∀𝑐 ∈ (Base‘𝑀)(𝑎(le‘𝑀)𝑏 → (𝑎(+g‘𝑀)𝑐)(le‘𝑀)(𝑏(+g‘𝑀)𝑐))))
5	4	simp2bi 1164	1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  +_gcplusg 17435  lecple 17442  Tosetctos 18595  Mndcmnd 18930  oMndcomnd 20340

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6496  df-fv 6549  df-ov 7425  df-omnd 20342

This theorem is used by:  omndadd2d  20351  omndadd2rd  20352  submomnd  20353  omndmul2  20354  omndmul  20356  gsumle  20366  orngsqr  21130  ofldtos  21137  isarchi3  33796  archirng  33797  archirngz  33798  archiabllem1a  33800  archiabllem1b  33801  archiabllem2a  33803  archiabllem2c  33804  archiabllem2b  33805  archiabl  33807