English
A ball around some center absorbs a smaller ball around the same center.
Русский
Шар вокруг некоторого центра всасывает меньший шар вокруг того же центра.
LaTeX
$$Absorbent 𝕜 (ball p x r)$$
Lean4
@[simp]
theorem smul_ball_preimage (p : Seminorm 𝕜 E) (y : E) (r : ℝ) (a : 𝕜) (ha : a ≠ 0) :
(a • ·) ⁻¹' p.ball y r = p.ball (a⁻¹ • y) (r / ‖a‖) :=
Set.ext fun _ => by
rw [mem_preimage, mem_ball, mem_ball, lt_div_iff₀ (norm_pos_iff.mpr ha), mul_comm, ← map_smul_eq_mul p, smul_sub,
smul_inv_smul₀ ha]