`BookProof.ChapterSolidHarmonic.solidHarmonic_spherical_inc` {l μ : ℕ} (hμ : μ ≤ l) {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : 0 ≤ Real.sin θ) (φ : ℝ) : solidHarmonic u v e l μ...

Proved

BookProof.ChapterSolidHarmonic.solidHarmonic_spherical_inc

by leonardopedro · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonic.

BookProof.ChapterSolidHarmonic.solidHarmonic_spherical_inc {l μ : ℕ} (hμ : μ ≤ l) {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : 0 ≤ Real.sin θ) (φ : ℝ) : solidHarmonic u v e l μ (spherePt u v e r θ φ) = r ^ l * assocLegendre l μ (Real.cos θ) * Real.cos (μ * φ)

Formalization note: Lean 4 identifier BookProof.ChapterSolidHarmonic.solidHarmonic_spherical_inc.

Replaces the retired node BookProof.ChapterSolidHarmonic.solidHarmonic_spherical, whose stored statement omitted the he, hu, hue, huv, hv, hve hypothesis binder (dropped include).

Preamble

-- Generated from ChapterSolidHarmonic.lean — theorem BookProof.ChapterSolidHarmonic.solidHarmonic_spherical_inc
import Definitions.Def_ChapterRadialLaplacian
import Definitions.Def_ChapterLaplacianProduct
import Definitions.Def_ChapterSolidHarmonicTools
import Mathlib
import Definitions.Def_ChapterSolidHarmonic
import Definitions.Def_ChapterLegendrePolynomial
open BookProof.ChapterLegendrePolynomial
open BookProof.ChapterSolidHarmonic



open Laplacian InnerProductSpace Polynomial
open BookProof.ChapterRadialLaplacian BookProof.ChapterLaplacianProduct
open BookProof.ChapterSolidHarmonicTools BookProof.ChapterLegendrePolynomial
open scoped RealInnerProductSpace

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

variable {u v e : E} (hu : ‖u‖ = 1) (hv : ‖v‖ = 1) (he : ‖e‖ = 1)
  (huv : ⟪u, v⟫_ℝ = 0) (hue : ⟪u, e⟫_ℝ = 0) (hve : ⟪v, e⟫_ℝ = 0)

include he hu hue huv hv hve in

Formal statement

theorem BookProof.ChapterSolidHarmonic.solidHarmonic_spherical_inc {l μ : ℕ} (hμ : μ ≤ l) {r : ℝ} (hr : 0 < r) {θ : ℝ}
    (hθ : 0 ≤ Real.sin θ) (φ : ℝ) :
    solidHarmonic u v e l μ (spherePt u v e r θ φ)
      = r ^ l * assocLegendre l μ (Real.cos θ) * Real.cos (μ * φ) := by sorry

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