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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlcvrp | Structured version Visualization version GIF version |
Description: A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 28618 analog.) (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlcvrp.b | ⊢ 𝐵 = (Base‘𝐾) |
cvlcvrp.j | ⊢ ∨ = (join‘𝐾) |
cvlcvrp.m | ⊢ ∧ = (meet‘𝐾) |
cvlcvrp.z | ⊢ 0 = (0.‘𝐾) |
cvlcvrp.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
cvlcvrp.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
cvlcvrp | ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp13 1086 | . . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ CvLat) | |
2 | cvllat 33631 | . . . . 5 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Lat) |
4 | simp2 1055 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
5 | cvlcvrp.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
6 | cvlcvrp.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 5, 6 | atbase 33594 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
8 | 7 | 3ad2ant3 1077 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐵) |
9 | cvlcvrp.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
10 | 5, 9 | latmcom 16898 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋)) |
11 | 3, 4, 8, 10 | syl3anc 1318 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋)) |
12 | 11 | eqeq1d 2612 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ (𝑃 ∧ 𝑋) = 0 )) |
13 | cvlatl 33630 | . . . 4 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
14 | 1, 13 | syl 17 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ AtLat) |
15 | simp3 1056 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐴) | |
16 | eqid 2610 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
17 | cvlcvrp.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
18 | 5, 16, 9, 17, 6 | atnle 33622 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃(le‘𝐾)𝑋 ↔ (𝑃 ∧ 𝑋) = 0 )) |
19 | 14, 15, 4, 18 | syl3anc 1318 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑋 ↔ (𝑃 ∧ 𝑋) = 0 )) |
20 | cvlcvrp.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
21 | cvlcvrp.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
22 | 5, 16, 20, 21, 6 | cvlcvr1 33644 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
23 | 12, 19, 22 | 3bitr2d 295 | 1 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 meetcmee 16768 0.cp0 16860 Latclat 16868 CLatccla 16930 OMLcoml 33480 ⋖ ccvr 33567 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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 |
This theorem is referenced by: cvlatcvr1 33646 cvrp 33720 |
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