Theorem omndmnd 29704

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.

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	simp1bi 1076	1 |- ( M e. oMnd -> M e. Mnd )

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  omndmul2  29712  omndmul3  29713  omndmul  29714  ogrpinv0le  29716  archirng  29742  gsumle  29779