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

Proved

BookProof.ChapterLaplacianProduct.fderiv_mul_eventually

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLaplacianProduct.

BookProof.ChapterLaplacianProduct.fderiv_mul_eventually {f g : E → ℝ} {x : E} (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) : (fderiv ℝ fun y : E => f y * g y) =ᶠ[nhds x] fun y => g y • fderiv ℝ f y + f y • fderiv ℝ g y

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

Preamble

-- Generated from ChapterLaplacianProduct.lean — theorem BookProof.ChapterLaplacianProduct.fderiv_mul_eventually
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.fderiv_mul_eventually {f g : E → ℝ} {x : E}
    (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) :
    (fderiv ℝ fun y : E => f y * g y) =ᶠ[nhds x]
      fun y => g y • fderiv ℝ f y + f y • fderiv ℝ g y := by sorry

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