Theorems · Theorem · real analysis

fderivWithin_multiset_prod

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {s : Set E} {ι : Type u_5} {𝔸' : Type u_7} [inst_3 : NormedCommRing 𝔸']
  [inst_4 : NormedAlgebra 𝕜 𝔸'] {g : ι → E → 𝔸'} [inst_5 : DecidableEq ι] {u : Multiset ι} {x : E},
  UniqueDiffWithinAt 𝕜 s x →
    (∀ i ∈ u, DifferentiableWithinAt 𝕜 (fun x => g i x) s x) →
      fderivWithin 𝕜 (fun x => (Multiset.map (fun x_1 => g x_1 x) u).prod) s x =
        (Multiset.map (fun i => (Multiset.map (fun x_1 => g x_1 x) (u.erase i)).prod • fderivWithin 𝕜 (g i) s x) u).sum

Defined in

Mathlib.Analysis.Calculus.FDeriv.Mul

Cited by

0 results in Mathlib

Foundations

Depth 187 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedCommRingNormedAlgebraDecidableEq