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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrp | 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 32063 analog.) (Contributed by NM, 18-Nov-2011.) |
Ref | Expression |
---|---|
cvrp.b | ⊢ 𝐵 = (Base‘𝐾) |
cvrp.j | ⊢ ∨ = (join‘𝐾) |
cvrp.m | ⊢ ∧ = (meet‘𝐾) |
cvrp.z | ⊢ 0 = (0.‘𝐾) |
cvrp.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
cvrp.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
cvrp | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlomcmcv 38693 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) | |
2 | cvrp.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | cvrp.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | cvrp.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
5 | cvrp.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
6 | cvrp.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
7 | cvrp.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 2, 3, 4, 5, 6, 7 | cvlcvrp 38677 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
9 | 1, 8 | syl3an1 1162 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 0 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 joincjn 18274 meetcmee 18275 0.cp0 18386 CLatccla 18461 OMLcoml 38512 ⋖ ccvr 38599 Atomscatm 38600 CvLatclc 38602 HLchlt 38687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-lat 18395 df-clat 18462 df-oposet 38513 df-ol 38515 df-oml 38516 df-covers 38603 df-ats 38604 df-atl 38635 df-cvlat 38659 df-hlat 38688 |
This theorem is referenced by: atcvrj1 38769 |
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