Theorems · Theorem · real analysis

ContDiffAt.smul

∀ {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] {x : E}
  {n : WithTop ℕ∞} {𝕜' : Type u_3} [inst_5 : NormedRing 𝕜'] [inst_6 : NormedAlgebra 𝕜 𝕜'] [inst_7 : Module 𝕜' F]
  [IsBoundedSMul 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {f : E → 𝕜'} {g : E → F},
  ContDiffAt 𝕜 n f x → ContDiffAt 𝕜 n g x → ContDiffAt 𝕜 n (f • g) x

The scalar multiplication of two C^n functions at a point is C^n at this point.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Operations

Cited by

2 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedRingNormedAlgebraModuleIsBoundedSMulIsScalarTower