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Definition df-llines 33601
Description: Define the set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice 𝑘, in other words all elements of height 2 (or lattice dimension 2 or projective dimension 1). (Contributed by NM, 16-Jun-2012.)
Assertion
Ref Expression
df-llines LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
Distinct variable group:   𝑘,𝑝,𝑥

Detailed syntax breakdown of Definition df-llines
StepHypRef Expression
1 clln 33594 . 2 class LLines
2 vk . . 3 setvar 𝑘
3 cvv 3168 . . 3 class V
4 vp . . . . . . 7 setvar 𝑝
54cv 1473 . . . . . 6 class 𝑝
6 vx . . . . . . 7 setvar 𝑥
76cv 1473 . . . . . 6 class 𝑥
82cv 1473 . . . . . . 7 class 𝑘
9 ccvr 33366 . . . . . . 7 class
108, 9cfv 5786 . . . . . 6 class ( ⋖ ‘𝑘)
115, 7, 10wbr 4573 . . . . 5 wff 𝑝( ⋖ ‘𝑘)𝑥
12 catm 33367 . . . . . 6 class Atoms
138, 12cfv 5786 . . . . 5 class (Atoms‘𝑘)
1411, 4, 13wrex 2892 . . . 4 wff 𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥
15 cbs 15637 . . . . 5 class Base
168, 15cfv 5786 . . . 4 class (Base‘𝑘)
1714, 6, 16crab 2895 . . 3 class {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥}
182, 3, 17cmpt 4633 . 2 class (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
191, 18wceq 1474 1 wff LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
Colors of variables: wff setvar class
This definition is referenced by:  llnset  33608
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