completeSpace_iff — Mathlib

English

If α and β are related by an isometry e: α ≃ᵢ β, then α is complete if and only if β is complete.

Русский

Если пространства α и β связаны изометрией e: α ≃ᵢ β, то α полното равно β полноте.

LaTeX

$$$\text{CompleteSpace}(\alpha) \iff \text{CompleteSpace}(\beta) \text{ when } e: \alpha \simeq_i \beta$$$

Lean4

theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpace β := by
  simp only [completeSpace_iff_isComplete_univ, ← e.range_eq_univ, ← image_univ,
    isComplete_image_iff e.isometry.isUniformInducing]

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