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Definition df-pclN 38754
Description: Projective subspace closure, which is the smallest projective subspace containing an arbitrary set of atoms. The subspace closure of the union of a set of projective subspaces is their supremum in PSubSp. Related to an analogous definition of closure used in Lemma 3.1.4 of [PtakPulmannova] p. 68. (Note that this closure is not necessarily one of the closed projective subspaces PSubCl of df-psubclN 38801.) (Contributed by NM, 7-Sep-2013.)
Assertion
Ref Expression
df-pclN PCl = (π‘˜ ∈ V ↦ (π‘₯ ∈ 𝒫 (Atomsβ€˜π‘˜) ↦ ∩ {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦}))
Distinct variable group:   π‘₯,π‘˜,𝑦

Detailed syntax breakdown of Definition df-pclN
StepHypRef Expression
1 cpclN 38753 . 2 class PCl
2 vk . . 3 setvar π‘˜
3 cvv 3474 . . 3 class V
4 vx . . . 4 setvar π‘₯
52cv 1540 . . . . . 6 class π‘˜
6 catm 38128 . . . . . 6 class Atoms
75, 6cfv 6543 . . . . 5 class (Atomsβ€˜π‘˜)
87cpw 4602 . . . 4 class 𝒫 (Atomsβ€˜π‘˜)
94cv 1540 . . . . . . 7 class π‘₯
10 vy . . . . . . . 8 setvar 𝑦
1110cv 1540 . . . . . . 7 class 𝑦
129, 11wss 3948 . . . . . 6 wff π‘₯ βŠ† 𝑦
13 cpsubsp 38362 . . . . . . 7 class PSubSp
145, 13cfv 6543 . . . . . 6 class (PSubSpβ€˜π‘˜)
1512, 10, 14crab 3432 . . . . 5 class {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦}
1615cint 4950 . . . 4 class ∩ {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦}
174, 8, 16cmpt 5231 . . 3 class (π‘₯ ∈ 𝒫 (Atomsβ€˜π‘˜) ↦ ∩ {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦})
182, 3, 17cmpt 5231 . 2 class (π‘˜ ∈ V ↦ (π‘₯ ∈ 𝒫 (Atomsβ€˜π‘˜) ↦ ∩ {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦}))
191, 18wceq 1541 1 wff PCl = (π‘˜ ∈ V ↦ (π‘₯ ∈ 𝒫 (Atomsβ€˜π‘˜) ↦ ∩ {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦}))
Colors of variables: wff setvar class
This definition is referenced by:  pclfvalN  38755
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