Theorems · Theorem · real analysis

ContDiffAt.prodMap

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {n : WithTop ℕ∞}
  {E' : Type u_3} [inst_5 : NormedAddCommGroup E'] [inst_6 : NormedSpace 𝕜 E'] {F' : Type u_4}
  [inst_7 : NormedAddCommGroup F'] [inst_8 : NormedSpace 𝕜 F'] {f : E → F} {g : E' → F'} {x : E} {y : E'},
  ContDiffAt 𝕜 n f x → ContDiffAt 𝕜 n g y → ContDiffAt 𝕜 n (Prod.map f g) (x, y)

The product map of two C^n functions within a set at a point is C^n within the product set at the product point.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Operations

Cited by

2 results in Mathlib

Foundations

Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace