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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 2798 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2798 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2798 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2798 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | iscvlat 36619 | . 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 501 | 1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 joincjn 17546 Atomscatm 36559 AtLatcal 36560 CvLatclc 36561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-cvlat 36618 |
This theorem is referenced by: cvllat 36622 cvlexch3 36628 cvlexch4N 36629 cvlatexchb1 36630 cvlcvr1 36635 cvlcvrp 36636 cvlatcvr1 36637 cvlsupr2 36639 hlatl 36656 |
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