Theorems · Theorem · real analysis

iteratedDerivWithin_scomp_eq_sum_orderedFinpartition

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {g : 𝕜 → E} {f : 𝕜 → 𝕜} {s t : Set 𝕜} {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, (∏ j, iteratedDerivWithin (c.partSize j) f s x) • iteratedDerivWithin c.length g t (f x)

Defined in

Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

Cited by

1 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace