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 33065
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 2735 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2735 . . 3 (+g𝑀) = (+g𝑀)
3 eqid 2735 . . 3 (le‘𝑀) = (le‘𝑀)
41, 2, 3isomnd 33061 . 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 1145 1 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wral 3059   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  lecple 17305  Tosetctos 18474  Mndcmnd 18760  oMndcomnd 33057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-omnd 33059
This theorem is referenced by:  omndadd2d  33068  omndadd2rd  33069  submomnd  33070  omndmul2  33072  omndmul  33074  gsumle  33084  isarchi3  33177  archirng  33178  archirngz  33179  archiabllem1a  33181  archiabllem1b  33182  archiabllem2a  33184  archiabllem2c  33185  archiabllem2b  33186  archiabl  33188  orngsqr  33314  ofldtos  33321
  Copyright terms: Public domain W3C validator