Theorems · Theorem · real analysis

iteratedDerivWithin_vcomp_eq_sum_orderedFinpartition

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {g : E → F} {f : 𝕜 → E}
  {s : Set 𝕜} {t : Set E} {x : 𝕜} {n : WithTop ℕ∞} {i : ℕ},
  ContDiffWithinAt 𝕜 n g t (f x) →
    ContDiffWithinAt 𝕜 n f s x →
      UniqueDiffOn 𝕜 t →
        UniqueDiffOn 𝕜 s →
          x ∈ s →
            Set.MapsTo f s t →
              ↑i ≤ n →
                iteratedDerivWithin i (g ∘ f) s x =
                  ∑ c, (iteratedFDerivWithin 𝕜 c.length g t (f x)) fun j => iteratedDerivWithin (c.partSize j) f s x

Defined in

Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

Cited by

4 results in Mathlib

Foundations

Depth 194 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace