Theorems · Theorem · real analysis

iteratedFDerivWithin_comp_of_eventually_mem

∀ {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] {s : Set E} {f : E → F} {g : F → G} {x : E}
  {n : WithTop ℕ∞} {t : Set F},
  ContDiffWithinAt 𝕜 n g t (f x) →
    ContDiffWithinAt 𝕜 n f s x →
      UniqueDiffOn 𝕜 t →
        UniqueDiffOn 𝕜 s →
          x ∈ s →
            (∀ᶠ (y : E) in nhdsWithin x s, f y ∈ t) →
              ∀ {i : ℕ},
                ↑i ≤ n →
                  iteratedFDerivWithin 𝕜 i (g ∘ f) s x =
                    (ftaylorSeriesWithin 𝕜 g t (f x)).taylorComp (ftaylorSeriesWithin 𝕜 f s x) i

Defined in

Mathlib.Analysis.Calculus.ContDiff.Comp

Cited by

1 results in Mathlib

Foundations

Depth 192 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace