Theorem omndtos 20221

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 2765	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2765	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2765	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 20217	. 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 2146  ∀wral 3081   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  +_gcplusg 17328  lecple 17335  Tosetctos 18488  Mndcmnd 18814  oMndcomnd 20213

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7419  df-omnd 20215

This theorem is used by:  omndadd2d  20224  omndadd2rd  20225  submomnd  20226  omndmul2  20227  omndmul  20229  gsumle  20239  orngsqr  20999  ofldtos  21006  isarchi3  33547  archirng  33548  archirngz  33549  archiabllem1a  33551  archiabllem1b  33552  archiabllem2a  33554  archiabllem2c  33555  archiabllem2b  33556  archiabl  33558