Theorems · Theorem · real analysis

ContDiff.fderiv

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [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] {m n : WithTop ℕ∞} {f : E → F → G} {g : E → F},
  ContDiff 𝕜 m (Function.uncurry f) → ContDiff 𝕜 n g → n + 1 ≤ m → ContDiff 𝕜 n fun x => fderiv 𝕜 (f x) (g x)

x ↦ fderiv 𝕜 (f x) (g x) is smooth.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Comp

Cited by

3 results in Mathlib

Foundations

Depth 206 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace