`BookProof.ChapterSolidHarmonicTools.laplacian_sum` {ι : Type*} (s : Finset ι) (f : ι → E → ℝ) (x : E) (hf : ∀ i ∈ s, ContDiffAt ℝ 2 (f i) x) : (Δ fun y => ∑ i ∈ s, f i y) x...

Proved

BookProof.ChapterSolidHarmonicTools.laplacian_sum

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonicTools.

BookProof.ChapterSolidHarmonicTools.laplacian_sum {ι : Type*} (s : Finset ι) (f : ι → E → ℝ) (x : E) (hf : ∀ i ∈ s, ContDiffAt ℝ 2 (f i) x) : (Δ fun y => ∑ i ∈ s, f i y) x = ∑ i ∈ s, (Δ (f i)) x

Formalization note: Lean 4 identifier BookProof.ChapterSolidHarmonicTools.laplacian_sum.

Preamble

-- Generated from ChapterSolidHarmonicTools.lean — theorem BookProof.ChapterSolidHarmonicTools.laplacian_sum
import Definitions.Def_ChapterRadialLaplacian
import Definitions.Def_ChapterLaplacianProduct
import Mathlib
import Definitions.Def_ChapterSolidHarmonicTools
open BookProof.ChapterSolidHarmonicTools



open Laplacian InnerProductSpace BookProof.ChapterRadialLaplacian
open BookProof.ChapterLaplacianProduct
open scoped RealInnerProductSpace


variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]

Formal statement

theorem BookProof.ChapterSolidHarmonicTools.laplacian_sum {ι : Type*} (s : Finset ι) (f : ι → E → ℝ) (x : E)
    (hf : ∀ i ∈ s, ContDiffAt ℝ 2 (f i) x) :
    (Δ fun y => ∑ i ∈ s, f i y) x = ∑ i ∈ s, (Δ (f i)) x := by sorry

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