`BookProof.ChapterLaplacianProduct.laplacian_mul` [FiniteDimensional ℝ E] {f g : E → ℝ} {x : E} (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) : (Δ fun y : E => f y * g y) x =

Proved

BookProof.ChapterLaplacianProduct.laplacian_mul

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLaplacianProduct.

BookProof.ChapterLaplacianProduct.laplacian_mul [FiniteDimensional ℝ E] {f g : E → ℝ} {x : E} (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) : (Δ fun y : E => f y * g y) x = (Δ f) x * g x + f x * (Δ g) x + 2 * ∑ i, fderiv ℝ f x ((stdOrthonormalBasis ℝ E) i) * fderiv ℝ g x ((stdOrthonormalBasis ℝ E) i)

Formalization note: Lean 4 identifier BookProof.ChapterLaplacianProduct.laplacian_mul.

Preamble

-- Generated from ChapterLaplacianProduct.lean — theorem BookProof.ChapterLaplacianProduct.laplacian_mul
import Definitions.Def_ChapterRadialLaplacian
import Mathlib
import Definitions.Def_ChapterLaplacianProduct
open BookProof.ChapterLaplacianProduct



open Filter Laplacian InnerProductSpace BookProof.ChapterRadialLaplacian
open scoped InnerProductSpace RealInnerProductSpace

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

Formal statement

theorem BookProof.ChapterLaplacianProduct.laplacian_mul [FiniteDimensional ℝ E] {f g : E → ℝ} {x : E}
    (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) :
    (Δ fun y : E => f y * g y) x
      = (Δ f) x * g x + f x * (Δ g) x
        + 2 * ∑ i, fderiv ℝ f x ((stdOrthonormalBasis ℝ E) i)
            * fderiv ℝ g x ((stdOrthonormalBasis ℝ E) i) := by sorry

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