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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 2737 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 2 | eqid 2737 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 3 | eqid 2737 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
| 4 | eqid 2737 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
| 5 | 1, 2, 3, 4 | iscvlat 39324 | . 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 497 | 1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 joincjn 18357 Atomscatm 39264 AtLatcal 39265 CvLatclc 39266 | 
| 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 | 
| 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-ral 3062 df-rab 3437 df-v 3482 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-cvlat 39323 | 
| This theorem is referenced by: cvllat 39327 cvlexch3 39333 cvlexch4N 39334 cvlatexchb1 39335 cvlcvr1 39340 cvlcvrp 39341 cvlatcvr1 39342 cvlsupr2 39344 hlatl 39361 | 
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