Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omndtos Structured version   Visualization version   GIF version

Theorem omndtos 28837
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.
StepHypRef Expression
1 eqid 2604 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2604 . . 3 (+g𝑀) = (+g𝑀)
3 eqid 2604 . . 3 (le‘𝑀) = (le‘𝑀)
41, 2, 3isomnd 28833 . 2 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)∀𝑐 ∈ (Base‘𝑀)(𝑎(le‘𝑀)𝑏 → (𝑎(+g𝑀)𝑐)(le‘𝑀)(𝑏(+g𝑀)𝑐))))
54simp2bi 1069 1 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1975  wral 2890   class class class wbr 4572  cfv 5785  (class class class)co 6522  Basecbs 15636  +gcplusg 15709  lecple 15716  Tosetctos 16797  Mndcmnd 17058  oMndcomnd 28829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-nul 4707
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-iota 5749  df-fv 5793  df-ov 6525  df-omnd 28831
This theorem is referenced by:  omndadd2d  28840  omndadd2rd  28841  submomnd  28842  omndmul2  28844  omndmul  28846  isarchi3  28873  archirng  28874  archirngz  28875  archiabllem1a  28877  archiabllem1b  28878  archiabllem2a  28880  archiabllem2c  28881  archiabllem2b  28882  archiabl  28884  gsumle  28911  orngsqr  28936  ofldtos  28943
  Copyright terms: Public domain W3C validator