Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omndmnd Structured version   Unicode version

Theorem omndmnd 28305
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 2429 . . 3  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2429 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
3 eqid 2429 . . 3  |-  ( le
`  M )  =  ( le `  M
)
41, 2, 3isomnd 28302 . 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 1020 1  |-  ( M  e. oMnd  ->  M  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   A.wral 2782   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   lecple 15159  Tosetctos 16230   Mndcmnd 16486  oMndcomnd 28298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-nul 4556
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-iota 5565  df-fv 5609  df-ov 6308  df-omnd 28300
This theorem is referenced by:  omndadd2d  28309  omndadd2rd  28310  omndmul2  28313  omndmul3  28314  omndmul  28315  ogrpinv0le  28317  archirng  28343  gsumle  28380
  Copyright terms: Public domain W3C validator