Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 2610 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2610 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2610 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | iscvlat 33628 | . 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 475 | 1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 Atomscatm 33568 AtLatcal 33569 CvLatclc 33570 |
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 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 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-cvlat 33627 |
This theorem is referenced by: cvllat 33631 cvlexch3 33637 cvlexch4N 33638 cvlatexchb1 33639 cvlcvr1 33644 cvlcvrp 33645 cvlatcvr1 33646 cvlsupr2 33648 hlatl 33665 |
Copyright terms: Public domain | W3C validator |