mdifferentiable_sub_section — Mathlib

English

If a function f is differentiable and a section s is differentiable, then the product f · s is differentiable as a section.

Русский

Если функция f дифференцируема и секция s дифференцируема, то произведение f · s является дифференцируемой секцией.

LaTeX

$$$ MDifferentiableWithinAt I (I.prod 𝓘(𝕜, F)) (\x => TotalSpace.mk' F x (f x \cdot s x)) x0$ from differentiability of f and s$$

Lean4

theorem mdifferentiable_sub_section (hs : MDifferentiable I (I.prod 𝓘(𝕜, F)) (fun x ↦ TotalSpace.mk' F x (s x)))
    (ht : MDifferentiable I (I.prod 𝓘(𝕜, F)) (fun x ↦ TotalSpace.mk' F x (t x))) :
    MDifferentiable I (I.prod 𝓘(𝕜, F)) (fun x ↦ TotalSpace.mk' F x ((s - t) x)) := fun x₀ ↦
  mdifferentiableAt_sub_section (hs x₀) (ht x₀)

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