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Theorem omndmnd 32805
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 2728 . . 3 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2728 . . 3 (+gβ€˜π‘€) = (+gβ€˜π‘€)
3 eqid 2728 . . 3 (leβ€˜π‘€) = (leβ€˜π‘€)
41, 2, 3isomnd 32802 . 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 3058   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  lecple 17247  Tosetctos 18415  Mndcmnd 18701  oMndcomnd 32798
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 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-omnd 32800
This theorem is referenced by:  omndadd2d  32809  omndadd2rd  32810  omndmul2  32813  omndmul3  32814  omndmul  32815  ogrpinv0le  32816  gsumle  32825  archirng  32917
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