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Theorem omndtos 31962
Description: A left-ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)
Assertion
Ref Expression
omndtos (𝑀 ∈ oMnd β†’ 𝑀 ∈ Toset)

Proof of Theorem omndtos
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2733 . . 3 (+gβ€˜π‘€) = (+gβ€˜π‘€)
3 eqid 2733 . . 3 (leβ€˜π‘€) = (leβ€˜π‘€)
41, 2, 3isomnd 31958 . 2 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ βˆ€π‘Ž ∈ (Baseβ€˜π‘€)βˆ€π‘ ∈ (Baseβ€˜π‘€)βˆ€π‘ ∈ (Baseβ€˜π‘€)(π‘Ž(leβ€˜π‘€)𝑏 β†’ (π‘Ž(+gβ€˜π‘€)𝑐)(leβ€˜π‘€)(𝑏(+gβ€˜π‘€)𝑐))))
54simp2bi 1147 1 (𝑀 ∈ oMnd β†’ 𝑀 ∈ Toset)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  lecple 17145  Tosetctos 18310  Mndcmnd 18561  oMndcomnd 31954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-omnd 31956
This theorem is referenced by:  omndadd2d  31965  omndadd2rd  31966  submomnd  31967  omndmul2  31969  omndmul  31971  gsumle  31981  isarchi3  32072  archirng  32073  archirngz  32074  archiabllem1a  32076  archiabllem1b  32077  archiabllem2a  32079  archiabllem2c  32080  archiabllem2b  32081  archiabl  32083  orngsqr  32146  ofldtos  32153
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