Theorem omndmnd 20373

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 2763	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2763	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2763	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 20370	. 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 2147  ∀wral 3080   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  +_gcplusg 17460  lecple 17467  Tosetctos 18620  Mndcmnd 18955  oMndcomnd 20366

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5264

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-ov 7449  df-omnd 20368

This theorem is used by:  omndadd2d  20377  omndadd2rd  20378  omndmul2  20380  omndmul3  20381  omndmul  20382  ogrpinv0le  20383  gsumle  20392  archirng  33825