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

Proved

BookProof.ChapterSolidHarmonicTools.hasFDerivAt_cylTerm

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonicTools.

BookProof.ChapterSolidHarmonicTools.hasFDerivAt_cylTerm (e : E) (j m : ℕ) (x : E) : HasFDerivAt (fun y : E => (⟪e, y⟫_ℝ) ^ j * (‖y‖ ^ 2) ^ m) (((j : ℝ) * (⟪e, x⟫_ℝ) ^ (j - 1) * (‖x‖ ^ 2) ^ m) • innerCLM E e + (2 * m * (⟪e, x⟫_ℝ) ^ j * (‖x‖ ^ 2) ^ (m - 1)) • innerCLM E x) x

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

Preamble

-- Generated from ChapterSolidHarmonicTools.lean — theorem BookProof.ChapterSolidHarmonicTools.hasFDerivAt_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.hasFDerivAt_cylTerm (e : E) (j m : ℕ) (x : E) :
    HasFDerivAt (fun y : E => (⟪e, y⟫_ℝ) ^ j * (‖y‖ ^ 2) ^ m)
      (((j : ℝ) * (⟪e, x⟫_ℝ) ^ (j - 1) * (‖x‖ ^ 2) ^ m) • innerCLM E e
        + (2 * m * (⟪e, x⟫_ℝ) ^ j * (‖x‖ ^ 2) ^ (m - 1)) • innerCLM E x) x := by sorry

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