Theorems · Theorem · global analysis

fderivWithin_clm_comp

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {G : Type u_4}
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {x : E} {s : Set E} {H : Type u_5}
  [inst_7 : NormedAddCommGroup H] [inst_8 : NormedSpace 𝕜 H] {c : E → G →L[𝕜] H} {d : E → F →L[𝕜] G},
  UniqueDiffWithinAt 𝕜 s x →
    DifferentiableWithinAt 𝕜 c s x →
      DifferentiableWithinAt 𝕜 d s x →
        fderivWithin 𝕜 (fun y => c y ∘SL d y) s x =
          (ContinuousLinearMap.compL 𝕜 F G H) (c x) ∘SL fderivWithin 𝕜 d s x +
            (ContinuousLinearMap.compL 𝕜 F G H).flip (d x) ∘SL fderivWithin 𝕜 c s x

Defined in

Mathlib.Analysis.Calculus.FDeriv.CompCLM

Cited by

0 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace