Theorems · Theorem · real analysis

ContDiffAt.prodMk

∀ {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] {x : E} {n : WithTop ℕ∞} {f : E → F} {g : E → G},
  ContDiffAt 𝕜 n f x → ContDiffAt 𝕜 n g x → ContDiffAt 𝕜 n (fun x => (f x, g x)) x

The Cartesian product of C^n functions at a point is C^n.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Basic

Cited by

8 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace