`BookProof.ChapterSolidHarmonicTools.laplacian_cylTerm` (e : E) (he : ‖e‖ = 1) (j m : ℕ) (x : E) : (Δ fun y : E => (⟪e, y⟫_ℝ) ^ j * (‖y‖ ^ 2) ^ m) x = (j : ℝ) * ((j : ℝ) - 1)...

Proved

BookProof.ChapterSolidHarmonicTools.laplacian_cylTerm

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonicTools.

BookProof.ChapterSolidHarmonicTools.laplacian_cylTerm (e : E) (he : ‖e‖ = 1) (j m : ℕ) (x : E) : (Δ fun y : E => (⟪e, y⟫_ℝ) ^ j * (‖y‖ ^ 2) ^ m) x = (j : ℝ) * ((j : ℝ) - 1) * (⟪e, x⟫_ℝ) ^ (j - 2) * (‖x‖ ^ 2) ^ m + (4 * m * ((m : ℝ) - 1) + 2 * (Module.finrank ℝ E) * m + 4 * j * m) * (⟪e, x⟫_ℝ) ^ j * (‖x‖ ^ 2) ^ (m - 1)

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

Preamble

-- Generated from ChapterSolidHarmonicTools.lean — theorem BookProof.ChapterSolidHarmonicTools.laplacian_cylTerm
import Definitions.Def_ChapterLaplacianProduct
import Mathlib
import Definitions.Def_ChapterSolidHarmonicTools
import Definitions.Def_ChapterRadialLaplacian
open BookProof.ChapterRadialLaplacian
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_cylTerm (e : E) (he : ‖e‖ = 1) (j m : ℕ) (x : E) :
    (Δ fun y : E => (⟪e, y⟫_ℝ) ^ j * (‖y‖ ^ 2) ^ m) x
      = (j : ℝ) * ((j : ℝ) - 1) * (⟪e, x⟫_ℝ) ^ (j - 2) * (‖x‖ ^ 2) ^ m
        + (4 * m * ((m : ℝ) - 1) + 2 * (Module.finrank ℝ E) * m + 4 * j * m)
            * (⟪e, x⟫_ℝ) ^ j * (‖x‖ ^ 2) ^ (m - 1) := by sorry

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