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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 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2821 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2821 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | iscvlat 36453 | . 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 500 | 1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 lecple 16566 joincjn 17548 Atomscatm 36393 AtLatcal 36394 CvLatclc 36395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-cvlat 36452 |
This theorem is referenced by: cvllat 36456 cvlexch3 36462 cvlexch4N 36463 cvlatexchb1 36464 cvlcvr1 36469 cvlcvrp 36470 cvlatcvr1 36471 cvlsupr2 36473 hlatl 36490 |
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