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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 | ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2733 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | eqid 2733 | . . 3 ⊢ (le‘𝑀) = (le‘𝑀) | |
4 | 1, 2, 3 | isomnd 32197 | . 2 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)∀𝑐 ∈ (Base‘𝑀)(𝑎(le‘𝑀)𝑏 → (𝑎(+g‘𝑀)𝑐)(le‘𝑀)(𝑏(+g‘𝑀)𝑐)))) |
5 | 4 | simp2bi 1147 | 1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3062 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 +gcplusg 17193 lecple 17200 Tosetctos 18365 Mndcmnd 18621 oMndcomnd 32193 |
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 5305 |
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 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-ov 7407 df-omnd 32195 |
This theorem is referenced by: omndadd2d 32204 omndadd2rd 32205 submomnd 32206 omndmul2 32208 omndmul 32210 gsumle 32220 isarchi3 32311 archirng 32312 archirngz 32313 archiabllem1a 32315 archiabllem1b 32316 archiabllem2a 32318 archiabllem2c 32319 archiabllem2b 32320 archiabl 32322 orngsqr 32391 ofldtos 32398 |
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