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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 2825 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2825 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | eqid 2825 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
4 | eqid 2825 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | iscvlat 35398 | . 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 493 | 1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∈ wcel 2166 ∀wral 3117 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 lecple 16312 joincjn 17297 Atomscatm 35338 AtLatcal 35339 CvLatclc 35340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 df-cvlat 35397 |
This theorem is referenced by: cvllat 35401 cvlexch3 35407 cvlexch4N 35408 cvlatexchb1 35409 cvlcvr1 35414 cvlcvrp 35415 cvlatcvr1 35416 cvlsupr2 35418 hlatl 35435 |
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