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lemma proved tactic proof

secondDeriv_smul

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formal statement (Lean)

  92lemma secondDeriv_smul (f : (Fin 4 → ℝ) → ℝ) (c : ℝ) (μ ν : Fin 4)
  93    (x : Fin 4 → ℝ)
  94    (h1 : ∀ y, DifferentiableAt ℝ (fun t => f (coordRay y μ t)) 0)
  95    (h2 : DifferentiableAt ℝ (fun s => partialDeriv_v2 f μ (coordRay x ν s)) 0) :
  96  secondDeriv (fun y => c * f y) μ ν x = c * secondDeriv f μ ν x := by

proof body

Tactic-mode proof.

  97  unfold secondDeriv
  98  have h_partial : ∀ y, partialDeriv_v2 (fun z => c * f z) μ y = c * partialDeriv_v2 f μ y := by
  99    intro y
 100    exact partialDeriv_v2_smul f c μ y (h1 y)
 101  simp only [h_partial]
 102  exact deriv_const_mul c h2
 103
 104/-- Laplacian linearity (scalar multiplication). -/

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