Theorem omndtos 20068

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 20064	. 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 2114  ∀wral 3052   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  lecple 17196  Tosetctos 18349  Mndcmnd 18671  oMndcomnd 20060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-omnd 20062

This theorem is referenced by:  omndadd2d  20071  omndadd2rd  20072  submomnd  20073  omndmul2  20074  omndmul  20076  gsumle  20086  orngsqr  20811  ofldtos  20818  isarchi3  33281  archirng  33282  archirngz  33283  archiabllem1a  33285  archiabllem1b  33286  archiabllem2a  33288  archiabllem2c  33289  archiabllem2b  33290  archiabl  33292