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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 33078 | . 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 2108 ∀wral 3061 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 lecple 17304 Tosetctos 18461 Mndcmnd 18747 oMndcomnd 33074 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-omnd 33076 | 
| This theorem is referenced by: omndadd2d 33085 omndadd2rd 33086 submomnd 33087 omndmul2 33089 omndmul 33091 gsumle 33101 isarchi3 33194 archirng 33195 archirngz 33196 archiabllem1a 33198 archiabllem1b 33199 archiabllem2a 33201 archiabllem2c 33202 archiabllem2b 33203 archiabl 33205 orngsqr 33334 ofldtos 33341 | 
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