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Theorem omndmnd 30736
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 2822 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2822 . . 3 (+g𝑀) = (+g𝑀)
3 eqid 2822 . . 3 (le‘𝑀) = (le‘𝑀)
41, 2, 3isomnd 30733 . 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 1142 1 (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  wral 3130   class class class wbr 5042  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  lecple 16563  Tosetctos 17634  Mndcmnd 17902  oMndcomnd 30729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-omnd 30731
This theorem is referenced by:  omndadd2d  30740  omndadd2rd  30741  omndmul2  30744  omndmul3  30745  omndmul  30746  ogrpinv0le  30747  gsumle  30756  archirng  30848
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