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Theorem omndmnd 27453
 Description: A left ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
Assertion
Ref Expression
omndmnd oMnd

Proof of Theorem omndmnd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3
2 eqid 2467 . . 3
3 eqid 2467 . . 3
41, 2, 3isomnd 27450 . 2 oMnd Toset
54simp1bi 1011 1 oMnd
 Colors of variables: wff setvar class Syntax hints:   wi 4   wcel 1767  wral 2814   class class class wbr 4447  cfv 5588  (class class class)co 6285  cbs 14493   cplusg 14558  cple 14565  Tosetctos 15523  cmnd 15729  oMndcomnd 27446 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6288  df-omnd 27448 This theorem is referenced by:  omndadd2d  27457  omndadd2rd  27458  omndmul2  27461  omndmul3  27462  omndmul  27463  ogrpinv0le  27465  archirng  27491  gsumle  27530
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