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Theorem omndmnd 33081
Description: A left-ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
Assertion
Ref Expression
omndmnd (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)

Proof of Theorem omndmnd
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2737 . . 3 (+g𝑀) = (+g𝑀)
3 eqid 2737 . . 3 (le‘𝑀) = (le‘𝑀)
41, 2, 3isomnd 33078 . 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))))
54simp1bi 1146 1 (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
Colors of variables: wff setvar class
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  omndmul2  33089  omndmul3  33090  omndmul  33091  ogrpinv0le  33092  gsumle  33101  archirng  33195
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