smul_toSimpleFunc — Mathlib

English

Scalar multiplication commutes with passing to simple functions, almost everywhere: toSimpleFunc(k · f) equals k · toSimpleFunc(f) almost everywhere.

Русский

Умножение на скаляр commuting с переходом к простым функциям: toSimpleFunc(k·f) ≈ k·toSimpleFunc(f) почти всюду.

LaTeX

$$$ toSimpleFunc (k \\cdot f) =_{\\mu} k \\cdot (toSimpleFunc\\,f) $$$

Lean4

theorem smul_toSimpleFunc (k : 𝕜) (f : Lp.simpleFunc E p μ) : toSimpleFunc (k • f) =ᵐ[μ] k • ⇑(toSimpleFunc f) :=
  by
  filter_upwards [toSimpleFunc_eq_toFun (k • f), toSimpleFunc_eq_toFun f, Lp.coeFn_smul k (f : Lp E p μ)] with _
  simp only [Pi.smul_apply, coe_smul]
  repeat intro h; rw [h]

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