Theorems · Theorem · real analysis

iteratedDerivWithin_smul

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type u_2} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {n : ℕ} {x : 𝕜} {s : Set 𝕜},
  x ∈ s →
    UniqueDiffOn 𝕜 s →
      ∀ {𝔸 : Type u_5} [inst_3 : NormedRing 𝔸] [inst_4 : NormedAlgebra 𝕜 𝔸] [inst_5 : Module 𝔸 F] [IsBoundedSMul 𝔸 F]
        [IsScalarTower 𝕜 𝔸 F] {f : 𝕜 → 𝔸} {g : 𝕜 → F},
        ContDiffWithinAt 𝕜 (↑n) f s x →
          ContDiffWithinAt 𝕜 (↑n) g s x →
            iteratedDerivWithin n (f • g) s x =
              ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x

Defined in

Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Cited by

1 results in Mathlib

Foundations

Depth 207 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedRingNormedAlgebraModuleIsBoundedSMulIsScalarTower