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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 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 | simp1bi 1069 | 1 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd) |
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 omndmul2 29043 omndmul3 29044 omndmul 29045 ogrpinv0le 29047 archirng 29073 gsumle 29110 |
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