Theorem omndtos 33026

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 2730	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2730	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2730	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 33022	. 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 1146	1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2109  ∀wral 3045   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  lecple 17234  Tosetctos 18382  Mndcmnd 18668  oMndcomnd 33018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-omnd 33020

This theorem is referenced by:  omndadd2d  33029  omndadd2rd  33030  submomnd  33031  omndmul2  33033  omndmul  33035  gsumle  33045  isarchi3  33148  archirng  33149  archirngz  33150  archiabllem1a  33152  archiabllem1b  33153  archiabllem2a  33155  archiabllem2c  33156  archiabllem2b  33157  archiabl  33159  orngsqr  33289  ofldtos  33296