mulTSupport_mul — Mathlib

English

The tsupport of the product f(x)·g(x) is contained in the tsupport of g.

Русский

Опора произведения лежит внутри опоры f(x)·g(x), но на слева.

LaTeX

$$$\operatorname{tsupport}(x \mapsto f(x) \cdot g(x)) \subseteq \operatorname{tsupport}(g)$$$

Lean4

@[to_additive]
theorem mulTSupport_mul [TopologicalSpace X] [MulOneClass α] {f g : X → α} :
    (mulTSupport fun x ↦ f x * g x) ⊆ mulTSupport f ∪ mulTSupport g :=
  closure_minimal ((mulSupport_mul f g).trans (union_subset_union (subset_mulTSupport _) (subset_mulTSupport _)))
    (isClosed_closure.union isClosed_closure)

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