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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 30146 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 1201 | . . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ CvLat) | |
2 | cvllat 36456 | . . . . 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 1133 | . . . 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 36419 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
8 | 7 | 3ad2ant3 1131 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐵) |
9 | cvlcvrp.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
10 | 5, 9 | latmcom 17679 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋)) |
11 | 3, 4, 8, 10 | syl3anc 1367 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋)) |
12 | 11 | eqeq1d 2823 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ (𝑃 ∧ 𝑋) = 0 )) |
13 | cvlatl 36455 | . . . 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 1134 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐴) | |
16 | eqid 2821 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
17 | cvlcvrp.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
18 | 5, 16, 9, 17, 6 | atnle 36447 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃(le‘𝐾)𝑋 ↔ (𝑃 ∧ 𝑋) = 0 )) |
19 | 14, 15, 4, 18 | syl3anc 1367 | . 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 36469 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
23 | 12, 19, 22 | 3bitr2d 309 | 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 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 lecple 16566 joincjn 17548 meetcmee 17549 0.cp0 17641 Latclat 17649 CLatccla 17711 OMLcoml 36305 ⋖ ccvr 36392 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 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
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-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 |
This theorem is referenced by: cvlatcvr1 36471 cvrp 36546 |
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