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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 32250 | . 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 3062 class class class wbr 5149 βcfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 lecple 17204 Tosetctos 18369 Mndcmnd 18625 oMndcomnd 32246 |
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 5307 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 df-omnd 32248 |
This theorem is referenced by: omndadd2d 32257 omndadd2rd 32258 submomnd 32259 omndmul2 32261 omndmul 32263 gsumle 32273 isarchi3 32364 archirng 32365 archirngz 32366 archiabllem1a 32368 archiabllem1b 32369 archiabllem2a 32371 archiabllem2c 32372 archiabllem2b 32373 archiabl 32375 orngsqr 32453 ofldtos 32460 |
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