instNatSMul — Mathlib

English

There exists a natural ℕ-action on ZeroHom(M,N) whenever M has a zero and N is an AddMonoid; the action is given by (a • f)(m) = a • f(m) and preserves zeros.

Русский

Существет естественное действо ℕ на ZeroHom(M,N) при наличии нуля в M и AddMonoid в N; действие задано (a • f)(m) = a • f(m) и сохраняет ноль.

LaTeX

$$$\\operatorname{ZeroHom}(M,N)$ has a natural $\\mathbb{N}$-action defined by $(a\\cdot f)(m) = a\\cdot f(m)$ with $(a\\cdot f)(0)=0$.$$

Lean4

instance instNatSMul [Zero M] [AddMonoid N] : SMul ℕ (ZeroHom M N) where
  smul a
    f :=
    { toFun := a • f
      map_zero' := by simp }

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