Theorem omndmnd 20227

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 2766	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2766	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2766	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 20224	. 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 1163	1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  ∀wral 3082   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  lecple 17342  Tosetctos 18495  Mndcmnd 18821  oMndcomnd 20220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-omnd 20222

This theorem is used by:  omndadd2d  20231  omndadd2rd  20232  omndmul2  20234  omndmul3  20235  omndmul  20236  ogrpinv0le  20237  gsumle  20246  archirng  33539