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Theorem omndmnd 32725
Description: A left-ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
Assertion
Ref Expression
omndmnd (𝑀 ∈ oMnd β†’ 𝑀 ∈ Mnd)

Proof of Theorem omndmnd
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . 3 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2726 . . 3 (+gβ€˜π‘€) = (+gβ€˜π‘€)
3 eqid 2726 . . 3 (leβ€˜π‘€) = (leβ€˜π‘€)
41, 2, 3isomnd 32722 . 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))))
54simp1bi 1142 1 (𝑀 ∈ oMnd β†’ 𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098  βˆ€wral 3055   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  +gcplusg 17203  lecple 17210  Tosetctos 18378  Mndcmnd 18664  oMndcomnd 32718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-omnd 32720
This theorem is referenced by:  omndadd2d  32729  omndadd2rd  32730  omndmul2  32733  omndmul3  32734  omndmul  32735  ogrpinv0le  32736  gsumle  32745  archirng  32837
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