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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 2737 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2737 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | eqid 2737 | . . 3 ⊢ (le‘𝑀) = (le‘𝑀) | |
4 | 1, 2, 3 | isomnd 31046 | . 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 1148 | 1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∀wral 3061 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +gcplusg 16802 lecple 16809 Tosetctos 17922 Mndcmnd 18173 oMndcomnd 31042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-nul 5199 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-iota 6338 df-fv 6388 df-ov 7216 df-omnd 31044 |
This theorem is referenced by: omndadd2d 31053 omndadd2rd 31054 submomnd 31055 omndmul2 31057 omndmul 31059 gsumle 31069 isarchi3 31160 archirng 31161 archirngz 31162 archiabllem1a 31164 archiabllem1b 31165 archiabllem2a 31167 archiabllem2c 31168 archiabllem2b 31169 archiabl 31171 orngsqr 31222 ofldtos 31229 |
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