instMonoid — Mathlib

English

If Y is a Monoid, LocallyConstant X Y inherits a Monoid structure with pointwise multiplication and a constant-ones as the unit: (f g)(x) = f(x) g(x), 1(x) = 1_Y.

Русский

Если Y — моноид, LocallyConstant X Y получает моноидовую структуру по точкам: (f g)(x) = f(x) g(x), 1(x) = 1_Y.

LaTeX

$$$\forall f,g:\mathrm{LocallyConstant}(X,Y),\; (f\cdot g)(x) = f(x)g(x),\; 1_{\mathrm{LocConst}}(x) = 1_Y.$$$

Lean4

@[to_additive]
instance [Monoid Y] : Monoid (LocallyConstant X Y) :=
  Function.Injective.monoid DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) fun _ _ => rfl

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