coe_smul — Mathlib

English

For any c ∈ M and x ∈ X, the embedding commutes with the smul: coe'(c ∘ x) = c ∘ coe'(x).

Русский

При любом c ∈ M и x ∈ X вложение commuting с умножением: coe'(c • x) = c • coe'(x).

LaTeX

$$$\\operatorname{coe}'(c \\cdot x) = c \\cdot \\operatorname{coe}'(x)$$$

Lean4

@[to_additive (attr := simp, norm_cast)]
theorem coe_smul (c : M) (x : X) : (↑(c • x) : Completion X) = c • (x : Completion X) :=
  (map_coe (uniformContinuous_const_smul c) x).symm

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