Definition df-llines 33802

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

Step	Hyp	Ref	Expression
1		clln 33795	. 2 class LLines
2		vk	. . 3 setvar 𝑘
3		cvv 3173	. . 3 class V
4		vp	. . . . . . 7 setvar 𝑝
5	4	cv 1474	. . . . . 6 class 𝑝
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1474	. . . . . 6 class 𝑥
8	2	cv 1474	. . . . . . 7 class 𝑘
9		ccvr 33567	. . . . . . 7 class ⋖
10	8, 9	cfv 5804	. . . . . 6 class ( ⋖ ‘𝑘)
11	5, 7, 10	wbr 4583	. . . . 5 wff 𝑝( ⋖ ‘𝑘)𝑥
12		catm 33568	. . . . . 6 class Atoms
13	8, 12	cfv 5804	. . . . 5 class (Atoms‘𝑘)
14	11, 4, 13	wrex 2897	. . . 4 wff ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥
15		cbs 15695	. . . . 5 class Base
16	8, 15	cfv 5804	. . . 4 class (Base‘𝑘)
17	14, 6, 16	crab 2900	. . 3 class {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥}
18	2, 3, 17	cmpt 4643	. 2 class (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
19	1, 18	wceq 1475	1 wff LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})

This definition is referenced by:  llnset  33809