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Definition df-pointsN 33806
 Description: Define set of all projective points in a Hilbert lattice (actually in any set at all, for simplicity). A projective point is the singleton of a lattice atom. Definition 15.1 of [MaedaMaeda] p. 61. Note that item 1 in [Holland95] p. 222 defines a point as the atom itself, but this leads to a complicated subspace ordering that may be either membership or inclusion based on its arguments. (Contributed by NM, 2-Oct-2011.)
Assertion
Ref Expression
df-pointsN Points = (𝑘 ∈ V ↦ {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}})
Distinct variable group:   𝑘,𝑝,𝑞

Detailed syntax breakdown of Definition df-pointsN
StepHypRef Expression
1 cpointsN 33799 . 2 class Points
2 vk . . 3 setvar 𝑘
3 cvv 3173 . . 3 class V
4 vq . . . . . . 7 setvar 𝑞
54cv 1474 . . . . . 6 class 𝑞
6 vp . . . . . . . 8 setvar 𝑝
76cv 1474 . . . . . . 7 class 𝑝
87csn 4125 . . . . . 6 class {𝑝}
95, 8wceq 1475 . . . . 5 wff 𝑞 = {𝑝}
102cv 1474 . . . . . 6 class 𝑘
11 catm 33568 . . . . . 6 class Atoms
1210, 11cfv 5804 . . . . 5 class (Atoms‘𝑘)
139, 6, 12wrex 2897 . . . 4 wff 𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}
1413, 4cab 2596 . . 3 class {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}}
152, 3, 14cmpt 4643 . 2 class (𝑘 ∈ V ↦ {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}})
161, 15wceq 1475 1 wff Points = (𝑘 ∈ V ↦ {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}})
 Colors of variables: wff setvar class This definition is referenced by:  pointsetN  34045
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