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Definition df-pointsN 37994
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 = (π‘˜ ∈ V ↦ {π‘ž ∣ βˆƒπ‘ ∈ (Atomsβ€˜π‘˜)π‘ž = {𝑝}})
Distinct variable group:   π‘˜,𝑝,π‘ž

Detailed syntax breakdown of Definition df-pointsN
StepHypRef Expression
1 cpointsN 37987 . 2 class Points
2 vk . . 3 setvar π‘˜
3 cvv 3448 . . 3 class V
4 vq . . . . . . 7 setvar π‘ž
54cv 1541 . . . . . 6 class π‘ž
6 vp . . . . . . . 8 setvar 𝑝
76cv 1541 . . . . . . 7 class 𝑝
87csn 4591 . . . . . 6 class {𝑝}
95, 8wceq 1542 . . . . 5 wff π‘ž = {𝑝}
102cv 1541 . . . . . 6 class π‘˜
11 catm 37754 . . . . . 6 class Atoms
1210, 11cfv 6501 . . . . 5 class (Atomsβ€˜π‘˜)
139, 6, 12wrex 3074 . . . 4 wff βˆƒπ‘ ∈ (Atomsβ€˜π‘˜)π‘ž = {𝑝}
1413, 4cab 2714 . . 3 class {π‘ž ∣ βˆƒπ‘ ∈ (Atomsβ€˜π‘˜)π‘ž = {𝑝}}
152, 3, 14cmpt 5193 . 2 class (π‘˜ ∈ V ↦ {π‘ž ∣ βˆƒπ‘ ∈ (Atomsβ€˜π‘˜)π‘ž = {𝑝}})
161, 15wceq 1542 1 wff Points = (π‘˜ ∈ V ↦ {π‘ž ∣ βˆƒπ‘ ∈ (Atomsβ€˜π‘˜)π‘ž = {𝑝}})
Colors of variables: wff setvar class
This definition is referenced by:  pointsetN  38233
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