Theorem omndtos 33082

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 2737	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2737	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2737	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 33078	. 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 1147	1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2108  ∀wral 3061   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  lecple 17304  Tosetctos 18461  Mndcmnd 18747  oMndcomnd 33074

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-omnd 33076

This theorem is referenced by:  omndadd2d  33085  omndadd2rd  33086  submomnd  33087  omndmul2  33089  omndmul  33091  gsumle  33101  isarchi3  33194  archirng  33195  archirngz  33196  archiabllem1a  33198  archiabllem1b  33199  archiabllem2a  33201  archiabllem2c  33202  archiabllem2b  33203  archiabl  33205  orngsqr  33334  ofldtos  33341