Theorem omndmnd 31232

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 2738	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2738	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2738	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 31229	. 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 1143	1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∈ wcel 2108  ∀wral 3063   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  lecple 16895  Tosetctos 18049  Mndcmnd 18300  oMndcomnd 31225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-omnd 31227

This theorem is referenced by:  omndadd2d  31236  omndadd2rd  31237  omndmul2  31240  omndmul3  31241  omndmul  31242  ogrpinv0le  31243  gsumle  31252  archirng  31344