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lemma proved tactic proof

secondDeriv_smul_local

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formal statement (Lean)

  77lemma secondDeriv_smul_local (f : (Fin 4 → ℝ) → ℝ) (c : ℝ) (μ ν : Fin 4)
  78    (x : Fin 4 → ℝ)
  79    (h1 : ∀ᶠ s in 𝓝 0, DifferentiableAt ℝ (fun t => f (coordRay (coordRay x ν s) μ t)) 0)
  80    (h2 : DifferentiableAt ℝ (fun s => partialDeriv_v2 f μ (coordRay x ν s)) 0) :
  81  secondDeriv (fun y => c * f y) μ ν x = c * secondDeriv f μ ν x := by

proof body

Tactic-mode proof.

  82  unfold secondDeriv
  83  have h_ev : ∀ᶠ s in 𝓝 0, partialDeriv_v2 (fun z => c * f z) μ (coordRay x ν s) =
  84                          c * partialDeriv_v2 f μ (coordRay x ν s) := by
  85    apply h1.mono
  86    intro s hs
  87    exact partialDeriv_v2_smul f c μ (coordRay x ν s) hs
  88  rw [Filter.EventuallyEq.deriv_eq h_ev]
  89  exact deriv_const_mul c h2
  90
  91/-- Second derivative linearity (scalar multiplication). -/

depends on (5)

Lean names referenced from this declaration's body.