Theorem omndmnd 20096

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 2741	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2741	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2741	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 20093	. 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 1152	1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)

Syntax hints:   → wi 4   ∈ wcel 2121  ∀wral 3055   class class class wbr 5075  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  +_gcplusg 17215  lecple 17222  Tosetctos 18375  Mndcmnd 18697  oMndcomnd 20089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5231

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-ov 7363  df-omnd 20091

This theorem is referenced by:  omndadd2d  20100  omndadd2rd  20101  omndmul2  20103  omndmul3  20104  omndmul  20105  ogrpinv0le  20106  gsumle  20115  archirng  33273