Theorem omndtos 20193

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 2769	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2769	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2769	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 20189	. 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 1162	1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2149  ∀wral 3085   class class class wbr 5110  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  +_gcplusg 17306  lecple 17313  Tosetctos 18466  Mndcmnd 18788  oMndcomnd 20185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-ov 7411  df-omnd 20187

This theorem is referenced by:  omndadd2d  20196  omndadd2rd  20197  submomnd  20198  omndmul2  20199  omndmul  20201  gsumle  20211  orngsqr  20943  ofldtos  20950  isarchi3  33444  archirng  33445  archirngz  33446  archiabllem1a  33448  archiabllem1b  33449  archiabllem2a  33451  archiabllem2c  33452  archiabllem2b  33453  archiabl  33455