Theorem omndtos 20334

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 2761	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2761	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2761	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 20330	. 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 2145  ∀wral 3077   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  +_gcplusg 17421  lecple 17428  Tosetctos 18581  Mndcmnd 18916  oMndcomnd 20326

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 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

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 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6493  df-fv 6545  df-ov 7421  df-omnd 20328

This theorem is used by:  omndadd2d  20337  omndadd2rd  20338  submomnd  20339  omndmul2  20340  omndmul  20342  gsumle  20352  orngsqr  21116  ofldtos  21123  isarchi3  33741  archirng  33742  archirngz  33743  archiabllem1a  33745  archiabllem1b  33746  archiabllem2a  33748  archiabllem2c  33749  archiabllem2b  33750  archiabl  33752