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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omndmnd | Structured version Visualization version GIF version |
Description: A left-ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
Ref | Expression |
---|---|
omndmnd | β’ (π β oMnd β π β Mnd) |
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 | simp1bi 1146 | 1 β’ (π β oMnd β π β Mnd) |
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 omndmul2 31969 omndmul3 31970 omndmul 31971 ogrpinv0le 31972 gsumle 31981 archirng 32073 |
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