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Theorem omndmnd 31961
Description: A left-ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
Assertion
Ref Expression
omndmnd (𝑀 ∈ oMnd β†’ 𝑀 ∈ Mnd)

Proof of Theorem omndmnd
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2733 . . 3 (+gβ€˜π‘€) = (+gβ€˜π‘€)
3 eqid 2733 . . 3 (leβ€˜π‘€) = (leβ€˜π‘€)
41, 2, 3isomnd 31958 . 2 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ βˆ€π‘Ž ∈ (Baseβ€˜π‘€)βˆ€π‘ ∈ (Baseβ€˜π‘€)βˆ€π‘ ∈ (Baseβ€˜π‘€)(π‘Ž(leβ€˜π‘€)𝑏 β†’ (π‘Ž(+gβ€˜π‘€)𝑐)(leβ€˜π‘€)(𝑏(+gβ€˜π‘€)𝑐))))
54simp1bi 1146 1 (𝑀 ∈ oMnd β†’ 𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  lecple 17145  Tosetctos 18310  Mndcmnd 18561  oMndcomnd 31954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-omnd 31956
This theorem is referenced by:  omndadd2d  31965  omndadd2rd  31966  omndmul2  31969  omndmul3  31970  omndmul  31971  ogrpinv0le  31972  gsumle  31981  archirng  32073
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