smul_section — Mathlib

English

If two differentiable at a point sections s and t, then their sum s + t is differentiable at that point.

Русский

Если две секции дифференцируемы в точке, то их сумма s + t дифференцируема в этой точке.

LaTeX

$$$ MDifferentiableAt I (I.prod 𝓘(𝕜, F)) (\x => TotalSpace.mk' F x (s x + t x)) x0$ from differentiability of s and t at x0$$

Lean4

theorem smul_section (hf : MDifferentiableAt I 𝓘(𝕜) f x₀)
    (hs : MDifferentiableAt I (I.prod 𝓘(𝕜, F)) (fun x ↦ TotalSpace.mk' F x (s x)) x₀) :
    MDifferentiableAt I (I.prod 𝓘(𝕜, F)) (fun x ↦ TotalSpace.mk' F x (f x • s x)) x₀ :=
  by
  rw [← mdifferentiableWithinAt_univ] at hs ⊢
  exact .smul_section hf hs

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