English
Scaling a ball by a nonzero c yields a ball with radius scaled by ||c||.
Русский
Умножение шара на ненулевой c даёт шар радиуса ||c||·r вокруг c·x.
LaTeX
$$$c \cdot \mathrm{ball}(x,r) = \mathrm{ball}(c\cdot x, \lVert c \rVert \cdot r).$$$
Lean4
theorem smul_sphere' {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) : c • sphere x r = sphere (c • x) (‖c‖ * r) :=
by
ext y
rw [mem_smul_set_iff_inv_smul_mem₀ hc]
conv_lhs => rw [← inv_smul_smul₀ hc x]
simp only [mem_sphere, dist_smul₀, norm_inv, ← div_eq_inv_mul, div_eq_iff (norm_pos_iff.2 hc).ne', mul_comm r]