Theorem omndmnd 30700

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.

Step	Hyp	Ref	Expression
1		eqid 2821	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2821	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2821	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 30697	. 2 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)∀𝑐 ∈ (Base‘𝑀)(𝑎(le‘𝑀)𝑏 → (𝑎(+g‘𝑀)𝑐)(le‘𝑀)(𝑏(+g‘𝑀)𝑐))))
5	4	simp1bi 1141	1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∈ wcel 2110  ∀wral 3138   class class class wbr 5058  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  +_gcplusg 16559  lecple 16566  Tosetctos 17637  Mndcmnd 17905  oMndcomnd 30693

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7153  df-omnd 30695

This theorem is referenced by:  omndadd2d  30704  omndadd2rd  30705  omndmul2  30708  omndmul3  30709  omndmul  30710  ogrpinv0le  30711  gsumle  30720  archirng  30812