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Theorem omndmnd 26332
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 2454 . . 3  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2454 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
3 eqid 2454 . . 3  |-  ( le
`  M )  =  ( le `  M
)
41, 2, 3isomnd 26329 . 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 1003 1  |-  ( M  e. oMnd  ->  M  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   A.wral 2799   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   +g cplusg 14360   lecple 14367  Tosetctos 15325   Mndcmnd 15531  oMndcomnd 26325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4532
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206  df-omnd 26327
This theorem is referenced by:  omndadd2d  26336  omndadd2rd  26337  omndmul2  26340  omndmul3  26341  omndmul  26342  ogrpinv0le  26344  archirng  26370  gsumle  26411
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