Theorems · Theorem · global analysis

fderivWithin_clm_apply

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {G : Type u_4} [inst_3 : NormedAddCommGroup G] [inst_4 : NormedSpace 𝕜 G] {x : E}
  {s : Set E} {H : Type u_5} [inst_5 : NormedAddCommGroup H] [inst_6 : NormedSpace 𝕜 H] {c : E → G →L[𝕜] H} {u : E → G},
  UniqueDiffWithinAt 𝕜 s x →
    DifferentiableWithinAt 𝕜 c s x →
      DifferentiableWithinAt 𝕜 u s x →
        fderivWithin 𝕜 (fun y => (c y) (u y)) s x = c x ∘SL fderivWithin 𝕜 u s x + (fderivWithin 𝕜 c s x).flip (u x)

Defined in

Mathlib.Analysis.Calculus.FDeriv.CompCLM

Cited by

4 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace