Theorems · Theorem · real analysis

hasFDerivAtFilter_finCons

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {n : ℕ} {F' : Fin n.succ → Type u_6}
  [inst_3 : (i : Fin n.succ) → NormedAddCommGroup (F' i)] [inst_4 : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)]
  {φ : E → F' 0} {φs : E → (i : Fin n) → F' i.succ} {φ' : E →L[𝕜] (i : Fin n.succ) → F' i} {l : Filter (E × E)},
  HasFDerivAtFilter (fun x => Fin.cons (φ x) (φs x)) φ' l ↔
    HasFDerivAtFilter φ (ContinuousLinearMap.proj 0 ∘SL φ') l ∧
      HasFDerivAtFilter φs (Pi.compRightL 𝕜 F' Fin.succ ∘SL φ') l

Defined in

Mathlib.Analysis.Calculus.FDeriv.Prod

Cited by

6 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace