`BookProof.ChapterSolidHarmonic.solidHarmonicIm_harmonic` {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,...

Proved

BookProof.ChapterSolidHarmonic.solidHarmonicIm_harmonic

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonic.

BookProof.ChapterSolidHarmonic.solidHarmonicIm_harmonic {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) (h3 : Module.finrank ℝ E = 3) {l μ : ℕ} (hμ : μ ≤ l) (x : E) : (Δ (solidHarmonicIm u v e l μ)) x = 0

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

Preamble

-- Generated from ChapterSolidHarmonic.lean — theorem BookProof.ChapterSolidHarmonic.solidHarmonicIm_harmonic
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]

Formal statement

theorem BookProof.ChapterSolidHarmonic.solidHarmonicIm_harmonic {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)
    (h3 : Module.finrank ℝ E = 3) {l μ : ℕ} (hμ : μ ≤ l) (x : E) :
    (Δ (solidHarmonicIm u v e l μ)) x = 0 := by sorry

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