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Theorem omndtos 28160
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 2404 . . 3  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2404 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
3 eqid 2404 . . 3  |-  ( le
`  M )  =  ( le `  M
)
41, 2, 3isomnd 28156 . 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 1015 1  |-  ( M  e. oMnd  ->  M  e. Toset )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1844   A.wral 2756   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   Basecbs 14843   +g cplusg 14911   lecple 14918  Tosetctos 15989   Mndcmnd 16245  oMndcomnd 28152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-nul 4527
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-iota 5535  df-fv 5579  df-ov 6283  df-omnd 28154
This theorem is referenced by:  omndadd2d  28163  omndadd2rd  28164  submomnd  28165  omndmul2  28167  omndmul  28169  isarchi3  28196  archirng  28197  archirngz  28198  archiabllem1a  28200  archiabllem1b  28201  archiabllem2a  28203  archiabllem2c  28204  archiabllem2b  28205  archiabl  28207  gsumle  28234  orngsqr  28260  ofldtos  28267
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