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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 2610 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2610 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
3 | eqid 2610 | . . 3 ⊢ (le‘𝑀) = (le‘𝑀) | |
4 | 1, 2, 3 | isomnd 29032 | . 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 1070 | 1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 lecple 15775 Tosetctos 16856 Mndcmnd 17117 oMndcomnd 29028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-omnd 29030 |
This theorem is referenced by: omndadd2d 29039 omndadd2rd 29040 submomnd 29041 omndmul2 29043 omndmul 29045 isarchi3 29072 archirng 29073 archirngz 29074 archiabllem1a 29076 archiabllem1b 29077 archiabllem2a 29079 archiabllem2c 29080 archiabllem2b 29081 archiabl 29083 gsumle 29110 orngsqr 29135 ofldtos 29142 |
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