Theorems · Theorem · functional analysis

fderiv_continuousAlternatingMap_apply

∀ {𝕜 : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} [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] {f : E → F [⋀^ι]→L[𝕜] G} {g : ι → E → F} {x : E}
  [inst_7 : Fintype ι] [inst_8 : DecidableEq ι],
  DifferentiableAt 𝕜 f x →
    (∀ (i : ι), DifferentiableAt 𝕜 (g i) x) →
      fderiv 𝕜 (fun x => (f x) fun x_1 => g x_1 x) x =
        (ContinuousAlternatingMap.apply 𝕜 F G fun x_1 => g x_1 x) ∘SL fderiv 𝕜 f x +
          ∑ i, (f x).toContinuousLinearMap (fun x_1 => g x_1 x) i ∘SL fderiv 𝕜 (g i) x

Defined in

Mathlib.Analysis.Calculus.FDeriv.ContinuousAlternatingMap

Cited by

1 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeDecidableEq