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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 29785 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 1266 | . . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ CvLat) | |
2 | cvllat 35400 | . . . . 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 1171 | . . . 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 35363 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
8 | 7 | 3ad2ant3 1169 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐵) |
9 | cvlcvrp.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
10 | 5, 9 | latmcom 17435 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋)) |
11 | 3, 4, 8, 10 | syl3anc 1494 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋)) |
12 | 11 | eqeq1d 2827 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ (𝑃 ∧ 𝑋) = 0 )) |
13 | cvlatl 35399 | . . . 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 1172 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐴) | |
16 | eqid 2825 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
17 | cvlcvrp.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
18 | 5, 16, 9, 17, 6 | atnle 35391 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃(le‘𝐾)𝑋 ↔ (𝑃 ∧ 𝑋) = 0 )) |
19 | 14, 15, 4, 18 | syl3anc 1494 | . 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 35413 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
23 | 12, 19, 22 | 3bitr2d 299 | 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 198 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 lecple 16319 joincjn 17304 meetcmee 17305 0.cp0 17397 Latclat 17405 CLatccla 17467 OMLcoml 35249 ⋖ ccvr 35336 Atomscatm 35337 AtLatcal 35338 CvLatclc 35339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-lat 17406 df-clat 17468 df-oposet 35250 df-ol 35252 df-oml 35253 df-covers 35340 df-ats 35341 df-atl 35372 df-cvlat 35396 |
This theorem is referenced by: cvlatcvr1 35415 cvrp 35490 |
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