image_sphere — Mathlib

English

For a linear isometry e, the property of being continuous on a set is preserved under composition with e when considering functions from a topological space.

Русский

Для линейной изометрии e свойство непрерывности на множестве сохраняется под композициями функций.

LaTeX

$$Iff (ContinuousOn (f ∘ e)) (ContinuousOn f)$$

Lean4

@[simp]
theorem image_sphere (x : E) (r : ℝ) : e '' Metric.sphere x r = Metric.sphere (e x) r :=
  e.toIsometryEquiv.image_sphere x r

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