Theorems · Theorem · real analysis

ContDiffWithinAt.fderivWithin_right_apply

∀ {𝕜 : Type u_1} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] [inst_3 : NormedAddCommGroup G] [inst_4 : NormedSpace 𝕜 G] {m n : WithTop ℕ∞} {f : F → G}
  {k : F → F} {s : Set F} {x₀ : F},
  ContDiffWithinAt 𝕜 n f s x₀ →
    ContDiffWithinAt 𝕜 m k s x₀ →
      UniqueDiffOn 𝕜 s → m + 1 ≤ n → x₀ ∈ s → ContDiffWithinAt 𝕜 m (fun x => (fderivWithin 𝕜 f s x) (k x)) s x₀

x ↦ fderivWithin 𝕜 f s x (k x) is smooth at x₀ within s.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Comp

Cited by

1 results in Mathlib

Foundations

Depth 206 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace