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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  |-  ( M  e. oMnd  ->  M  e.  Mnd )

Proof of Theorem omndmnd
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2467 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
3 eqid 2467 . . 3  |-  ( le
`  M )  =  ( le `  M
)
41, 2, 3isomnd 27450 . 2  |-  ( M  e. oMnd 
<->  ( M  e.  Mnd  /\  M  e. Toset  /\  A. a  e.  ( Base `  M
) A. b  e.  ( Base `  M
) A. c  e.  ( Base `  M
) ( a ( le `  M ) b  ->  ( a
( +g  `  M ) c ) ( le
`  M ) ( b ( +g  `  M
) c ) ) ) )
54simp1bi 1011 1  |-  ( M  e. oMnd  ->  M  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   A.wral 2814   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   +g cplusg 14558   lecple 14565  Tosetctos 15523   Mndcmnd 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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