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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlatl | Structured version Visualization version GIF version |
Description: An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlatl | ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2725 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2725 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2725 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | iscvlat 38925 | . 2 ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)∀𝑥 ∈ (Base‘𝐾)((¬ 𝑝(le‘𝐾)𝑥 ∧ 𝑝(le‘𝐾)(𝑥(join‘𝐾)𝑞)) → 𝑞(le‘𝐾)(𝑥(join‘𝐾)𝑝)))) |
6 | 5 | simplbi 496 | 1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2098 ∀wral 3050 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 lecple 17243 joincjn 18306 Atomscatm 38865 AtLatcal 38866 CvLatclc 38867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-cvlat 38924 |
This theorem is referenced by: cvllat 38928 cvlexch3 38934 cvlexch4N 38935 cvlatexchb1 38936 cvlcvr1 38941 cvlcvrp 38942 cvlatcvr1 38943 cvlsupr2 38945 hlatl 38962 |
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