Theorems · Theorem · real analysis

ContDiffWithinAt.comp_of_mem_nhdsWithin_image

∀ {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] {n : WithTop ℕ∞} {s : Set E} {t : Set F} {g : F → G}
  {f : E → F} (x : E),
  ContDiffWithinAt 𝕜 n g t (f x) →
    ContDiffWithinAt 𝕜 n f s x → t ∈ nhdsWithin (f x) (f '' s) → ContDiffWithinAt 𝕜 n (g ∘ f) s x

The composition of C^n functions at points in domains is C^n, with a weaker condition on s and t.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Comp

Cited by

2 results in Mathlib

Foundations

Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace