circleAverage_smul — Mathlib

English

If f1 and f2 agree on the circle, their circle averages are equal (congruence on sphere).

Русский

Если f1 и f2 согласуются на окружности, их средние совпадают (конгруэнтность на сфере).

LaTeX

$$$circleAverage\ f_1\ c\ R = circleAverage\ f_2\ c\ R\;\text{если }\;f_1=f_2\text{ на окружности}$$$

Lean4

/-- Circle averages commute with scalar multiplication. -/
theorem circleAverage_smul : circleAverage (a • f) c R = a • circleAverage f c R :=
  by
  unfold circleAverage
  have := SMulCommClass.symm ℝ 𝕜 E
  rw [smul_comm]
  simp [intervalIntegral.integral_smul]

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