Theorems · Theorem · real analysis

ContDiffAt.fderiv

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {x₀ : E} {m n : WithTop ℕ∞} {f : E → F → G} {g : E → F},
  ContDiffAt 𝕜 n (Function.uncurry f) (x₀, g x₀) →
    ContDiffAt 𝕜 m g x₀ → m + 1 ≤ n → ContDiffAt 𝕜 m (fun x => fderiv 𝕜 (f x) (g x)) x₀

x ↦ fderiv 𝕜 (f x) (g x) is smooth at x₀.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Comp

Cited by

6 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace