Theorem omndtos 29705

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` M ) = ( Base ` M )
2		eqid 2622	. . 3 |- ( +g ` M ) = ( +g ` M )
3		eqid 2622	. . 3 |- ( le ` M ) = ( le ` M )
4	1, 2, 3	isomnd 29701	. 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 ) ) ) )
5	4	simp2bi 1077	1 |- ( M e. oMnd -> M e. Toset )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   lecple 15948  Tosetctos 17033   Mndcmnd 17294  oMndcomnd 29697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-omnd 29699

This theorem is referenced by:  omndadd2d  29708  omndadd2rd  29709  submomnd  29710  omndmul2  29712  omndmul  29714  isarchi3  29741  archirng  29742  archirngz  29743  archiabllem1a  29745  archiabllem1b  29746  archiabllem2a  29748  archiabllem2c  29749  archiabllem2b  29750  archiabl  29752  gsumle  29779  orngsqr  29804  ofldtos  29811