Theorem omndtos 20392

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 2763	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2763	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2763	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 20388	. 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 2147  ∀wral 3080   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  lecple 17479  Tosetctos 18638  Mndcmnd 18973  oMndcomnd 20384

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454  df-omnd 20386

This theorem is used by:  omndadd2d  20395  omndadd2rd  20396  submomnd  20397  omndmul2  20398  omndmul  20400  gsumle  20410  orngsqr  21174  ofldtos  21181  isarchi3  33847  archirng  33848  archirngz  33849  archiabllem1a  33851  archiabllem1b  33852  archiabllem2a  33854  archiabllem2c  33855  archiabllem2b  33856  archiabl  33858