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Theorem omndmnd 29035
 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 2610 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2610 . . 3 (+g𝑀) = (+g𝑀)
3 eqid 2610 . . 3 (le‘𝑀) = (le‘𝑀)
41, 2, 3isomnd 29032 . 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 1069 1 (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  lecple 15775  Tosetctos 16856  Mndcmnd 17117  oMndcomnd 29028 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-omnd 29030 This theorem is referenced by:  omndadd2d  29039  omndadd2rd  29040  omndmul2  29043  omndmul3  29044  omndmul  29045  ogrpinv0le  29047  archirng  29073  gsumle  29110
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