`BookProof.ChapterSolidHarmonicTools.laplacian_angular_mul` {A B : E → ℝ} {x : E} {α β : ℝ} {e : E} {μ : ℕ} (hA : ContDiffAt ℝ 2 A x) (hB : ContDiffAt ℝ 2 B x) (hharm : (Δ A) x = 0

Proved

BookProof.ChapterSolidHarmonicTools.laplacian_angular_mul

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonicTools.

BookProof.ChapterSolidHarmonicTools.laplacian_angular_mul {A B : E → ℝ} {x : E} {α β : ℝ} {e : E} {μ : ℕ} (hA : ContDiffAt ℝ 2 A x) (hB : ContDiffAt ℝ 2 B x) (hharm : (Δ A) x = 0) (heuler : fderiv ℝ A x x = μ * A x) (haxis : fderiv ℝ A x e = 0) (hgrad : fderiv ℝ B x = α • innerCLM E e + β • innerCLM E x) : (Δ fun y : E => A y * B y) x = A x * (Δ B) x + 2 * β * (μ : ℝ) * A x

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

Preamble

-- Generated from ChapterSolidHarmonicTools.lean — theorem BookProof.ChapterSolidHarmonicTools.laplacian_angular_mul
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_angular_mul {A B : E → ℝ} {x : E} {α β : ℝ} {e : E} {μ : ℕ}
    (hA : ContDiffAt ℝ 2 A x) (hB : ContDiffAt ℝ 2 B x)
    (hharm : (Δ A) x = 0) (heuler : fderiv ℝ A x x = μ * A x) (haxis : fderiv ℝ A x e = 0)
    (hgrad : fderiv ℝ B x = α • innerCLM E e + β • innerCLM E x) :
    (Δ fun y : E => A y * B y) x = A x * (Δ B) x + 2 * β * (μ : ℝ) * A x := by sorry

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