Theorem omndtos 30708

Description: A left-ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)

Assertion

Ref	Expression
omndtos	⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)

Proof of Theorem omndtos

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2823	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2823	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
3		eqid 2823	. . 3 ⊢ (le‘𝑀) = (le‘𝑀)
4	1, 2, 3	isomnd 30704	. 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	simp2bi 1142	1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3140   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  +_gcplusg 16567  lecple 16574  Tosetctos 17645  Mndcmnd 17913  oMndcomnd 30700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161  df-omnd 30702

This theorem is referenced by:  omndadd2d  30711  omndadd2rd  30712  submomnd  30713  omndmul2  30715  omndmul  30717  gsumle  30727  isarchi3  30818  archirng  30819  archirngz  30820  archiabllem1a  30822  archiabllem1b  30823  archiabllem2a  30825  archiabllem2c  30826  archiabllem2b  30827  archiabl  30829  orngsqr  30879  ofldtos  30886