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Theorem omndmnd 33064
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 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))))
54simp1bi 1144 1 (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
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  omndmul2  33072  omndmul3  33073  omndmul  33074  ogrpinv0le  33075  gsumle  33084  archirng  33178
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