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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 | ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2735 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | eqid 2735 | . . 3 ⊢ (le‘𝑀) = (le‘𝑀) | |
4 | 1, 2, 3 | isomnd 33061 | . 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 1144 | 1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 lecple 17305 Tosetctos 18474 Mndcmnd 18760 oMndcomnd 33057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-omnd 33059 |
This theorem is referenced by: omndadd2d 33068 omndadd2rd 33069 omndmul2 33072 omndmul3 33073 omndmul 33074 ogrpinv0le 33075 gsumle 33084 archirng 33178 |
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