contMDiff_smul — Mathlib

English

Multiplication by a scalar is a smooth operation on a vector space: the map (λ, v) ↦ λ · v is ContMDiff.

Русский

Умножение на скаляль гладкая операция на векторном пространстве: (λ, v) ↦ λ·v гладко зависимо от входов.

LaTeX

$$$\forall (p)\, (p.fst) \in \mathbb k, (p.snd) \in V,\ ContMDiff ((\mathcal I).prod (\mathcal I)) (\mathcal I) ⊗ ⊤ (\lambda p: 𝕜 \times V \mapsto p.1 \cdot p.2) $$$

Lean4

/-- On any vector space, multiplication by a scalar is a smooth operation. -/
theorem contMDiff_smul : ContMDiff (𝓘(𝕜).prod 𝓘(𝕜, V)) 𝓘(𝕜, V) ⊤ fun p : 𝕜 × V => p.1 • p.2 :=
  contMDiff_iff.2 ⟨continuous_smul, fun _ _ => contDiff_smul.contDiffOn⟩

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