Definition df-lplanes 34785

Description: Define the set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice 𝑘, in other words all elements of height 3 (or lattice dimension 3 or projective dimension 2). (Contributed by NM, 16-Jun-2012.)

Assertion

Ref	Expression
df-lplanes	⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})

Distinct variable group:   𝑘,𝑝,𝑥

Detailed syntax breakdown of Definition df-lplanes

Step	Hyp	Ref	Expression
1		clpl 34778	. 2 class LPlanes
2		vk	. . 3 setvar 𝑘
3		cvv 3200	. . 3 class V
4		vp	. . . . . . 7 setvar 𝑝
5	4	cv 1482	. . . . . 6 class 𝑝
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1482	. . . . . 6 class 𝑥
8	2	cv 1482	. . . . . . 7 class 𝑘
9		ccvr 34549	. . . . . . 7 class ⋖
10	8, 9	cfv 5888	. . . . . 6 class ( ⋖ ‘𝑘)
11	5, 7, 10	wbr 4653	. . . . 5 wff 𝑝( ⋖ ‘𝑘)𝑥
12		clln 34777	. . . . . 6 class LLines
13	8, 12	cfv 5888	. . . . 5 class (LLines‘𝑘)
14	11, 4, 13	wrex 2913	. . . 4 wff ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥
15		cbs 15857	. . . . 5 class Base
16	8, 15	cfv 5888	. . . 4 class (Base‘𝑘)
17	14, 6, 16	crab 2916	. . 3 class {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥}
18	2, 3, 17	cmpt 4729	. 2 class (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
19	1, 18	wceq 1483	1 wff LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})

This definition is referenced by:  lplnset  34815