`BookProof.ChapterSolidHarmonic.inner_u_spherePt` (r θ φ : ℝ) : ⟪u, spherePt u v e r θ φ⟫_ℝ = r * Real.sin θ * Real.cos φ
DisprovedBookProof.ChapterSolidHarmonic.inner_u_spherePtProve the following Lean 4 theorem from ChapterSolidHarmonic.
BookProof.ChapterSolidHarmonic.inner_u_spherePt (r θ φ : ℝ) : ⟪u, spherePt u v e r θ φ⟫_ℝ = r * Real.sin θ * Real.cos φ
Formalization note: Lean 4 identifier BookProof.ChapterSolidHarmonic.inner_u_spherePt.
Retired 2026-10-30 — superseded by a corrected statement, not withdrawn. The exported target lost the frame hypotheses, which were section variables not occurring in the type, so it claimed the coordinate identity for arbitrary vectors , , .
Where the fix lives. In the source of record the hypothesis is an explicit binder of the declaration, so a re-export cannot lose it:
BookProof.ChapterSolidHarmonic.inner_u_spherePt_frames_explicitintimepiece/BookProof/HarmonicAnalysis/ChapterSolidHarmonic.lean(Lean 4.28 tree),BookProof.ChapterSolidHarmonic.inner_u_spherePt_frames_explicitintimepiece331/BookProof/ChapterSolidHarmonic.lean(Lean 4.33.1 tree).
It is published on the platform as BookProof.ChapterSolidHarmonic.inner_u_spherePt_frames_explicit. This record stays readable and importable for anything already citing it; its proof status is unchanged.
-- Generated from ChapterSolidHarmonic.lean — theorem BookProof.ChapterSolidHarmonic.inner_u_spherePt
import Definitions.Def_ChapterRadialLaplacian
import Definitions.Def_ChapterLaplacianProduct
import Definitions.Def_ChapterSolidHarmonicTools
import Definitions.Def_ChapterLegendrePolynomial
import Mathlib
import Definitions.Def_ChapterSolidHarmonic
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)theorem BookProof.ChapterSolidHarmonic.inner_u_spherePt (r θ φ : ℝ) :
⟪u, spherePt u v e r θ φ⟫_ℝ = r * Real.sin θ * Real.cos φ := by sorry