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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omndtos | Structured version Visualization version GIF version |
Description: A left-ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
Ref | Expression |
---|---|
omndtos | β’ (π β oMnd β π β Toset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2733 | . . 3 β’ (+gβπ) = (+gβπ) | |
3 | eqid 2733 | . . 3 β’ (leβπ) = (leβπ) | |
4 | 1, 2, 3 | isomnd 31958 | . 2 β’ (π β oMnd β (π β Mnd β§ π β Toset β§ βπ β (Baseβπ)βπ β (Baseβπ)βπ β (Baseβπ)(π(leβπ)π β (π(+gβπ)π)(leβπ)(π(+gβπ)π)))) |
5 | 4 | simp2bi 1147 | 1 β’ (π β oMnd β π β Toset) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 βwral 3061 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 lecple 17145 Tosetctos 18310 Mndcmnd 18561 oMndcomnd 31954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5264 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-omnd 31956 |
This theorem is referenced by: omndadd2d 31965 omndadd2rd 31966 submomnd 31967 omndmul2 31969 omndmul 31971 gsumle 31981 isarchi3 32072 archirng 32073 archirngz 32074 archiabllem1a 32076 archiabllem1b 32077 archiabllem2a 32079 archiabllem2c 32080 archiabllem2b 32081 archiabl 32083 orngsqr 32146 ofldtos 32153 |
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