| 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 | ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd) |
| 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 | simp1bi 1146 | 1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd) |
| 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 omndmul2 33089 omndmul3 33090 omndmul 33091 ogrpinv0le 33092 gsumle 33101 archirng 33195 |
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