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Theorem omndtos 27357
Description: A left ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)
Assertion
Ref Expression
omndtos  |-  ( M  e. oMnd  ->  M  e. Toset )

Proof of Theorem omndtos
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 27353 . 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 ) ) ) )
54simp2bi 1012 1  |-  ( M  e. oMnd  ->  M  e. Toset )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   A.wral 2814   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   lecple 14558  Tosetctos 15516   Mndcmnd 15722  oMndcomnd 27349
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 5549  df-fv 5594  df-ov 6285  df-omnd 27351
This theorem is referenced by:  omndadd2d  27360  omndadd2rd  27361  submomnd  27362  omndmul2  27364  omndmul  27366  isarchi3  27393  archirng  27394  archirngz  27395  archiabllem1a  27397  archiabllem1b  27398  archiabllem2a  27400  archiabllem2c  27401  archiabllem2b  27402  archiabl  27404  gsumle  27433  orngsqr  27457  ofldtos  27464
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