mul_left_of_commute — Mathlib

English

Locally finite of supports is equivalent to locally finite of their multiplicative supports.

Русский

Локальная конечность опор эквивалентна локальной конечности их мультиподдержек.

LaTeX

$$$\operatorname{LocallyFinite}(i \mapsto \operatorname{mulSupport}(f_i)) \iff \operatorname{LocallyFinite}(i \mapsto \operatorname{mulTSupport}(f_i)).$$$

Lean4

/-- If `a` and `b` commute and `b` is topologically nilpotent,
  then `a * b` is topologically nilpotent. -/
theorem mul_left_of_commute [IsLinearTopology R R] {a b : R} (hb : IsTopologicallyNilpotent b) (hab : Commute a b) :
    IsTopologicallyNilpotent (a * b) :=
  by
  simp_rw [IsTopologicallyNilpotent, hab.mul_pow]
  exact IsLinearTopology.tendsto_mul_zero_of_right _ _ hb

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