Theorems · Theorem · global analysis

fderiv_comp_fderivWithin

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {G : Type u_4}
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {f : E → F} (x : E) {s : Set E} {g : F → G},
  DifferentiableAt 𝕜 g (f x) →
    DifferentiableWithinAt 𝕜 f s x →
      UniqueDiffWithinAt 𝕜 s x → fderivWithin 𝕜 (g ∘ f) s x = fderiv 𝕜 g (f x) ∘SL fderivWithin 𝕜 f s x

Defined in

Mathlib.Analysis.Calculus.FDeriv.Comp

Cited by

2 results in Mathlib

Foundations

Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace