File: Mathlib/Topology/MetricSpace/Completion.lean

full_name: UniformSpace.Completion.instDist
code: /-- The distance on the completion is obtained by extending the distance on the original space,
by uniform continuity. -/
instance : Dist (Completion α) 
goal: Dist (UniformSpace.Completion α)

full_name: UniformSpace.Completion.uniformContinuous_dist
code: /-- The new distance is uniformly continuous. -/
protected theorem uniformContinuous_dist :
    UniformContinuous fun p : Completion α × Completion α ↦ dist p.1 p.2 
goal: UniformContinuous fun p => dist p.1 p.2

full_name: UniformSpace.Completion.continuous_dist
code: /-- The new distance is continuous. -/
protected theorem continuous_dist [TopologicalSpace β] {f g : β → Completion α} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun x ↦ dist (f x) (g x) 
goal: Continuous fun x => dist (f x) (g x)

full_name: UniformSpace.Completion.dist_eq
code: /-- The new distance is an extension of the original distance. -/
@[simp]
protected theorem dist_eq (x y : α) : dist (x : Completion α) y = dist x y 
goal: dist (↑α x) (↑α y) = dist x y

full_name: UniformSpace.Completion.dist_self
code: protected theorem dist_self (x : Completion α) : dist x x = 0 
goal: dist x x = 0

full_name: UniformSpace.Completion.dist_comm
code: protected theorem dist_comm (x y : Completion α) : dist x y = dist y x 
goal: dist x y = dist y x

full_name: UniformSpace.Completion.dist_triangle
code: protected theorem dist_triangle (x y z : Completion α) : dist x z ≤ dist x y + dist y z 
goal: dist x z ≤ dist x y + dist y z

full_name: UniformSpace.Completion.mem_uniformity_dist
code: /-- Elements of the uniformity (defined generally for completions) can be characterized in terms
of the distance. -/
protected theorem mem_uniformity_dist (s : Set (Completion α × Completion α)) :
    s ∈ 𝓤 (Completion α) ↔ ∃ ε > 0, ∀ {a b}, dist a b < ε → (a, b) ∈ s 
goal: s ∈ 𝓤 (UniformSpace.Completion α) ↔ ∃ ε > 0, ∀ {a b : UniformSpace.Completion α}, dist a b < ε → (a, b) ∈ s

full_name: UniformSpace.Completion.uniformity_dist'
code: /-- Reformulate `Completion.mem_uniformity_dist` in terms that are suitable for the definition
of the metric space structure. -/
protected theorem uniformity_dist' :
    𝓤 (Completion α) = ⨅ ε : { ε : ℝ // 0 < ε }, 𝓟 { p | dist p.1 p.2 < ε.val } 
goal: 𝓤 (UniformSpace.Completion α) = ⨅ ε, 𝓟 {p | dist p.1 p.2 < ↑ε}

full_name: UniformSpace.Completion.uniformity_dist
code: protected theorem uniformity_dist : 𝓤 (Completion α) = ⨅ ε > 0, 𝓟 { p | dist p.1 p.2 < ε } 
goal: 𝓤 (UniformSpace.Completion α) = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε}

full_name: UniformSpace.Completion.instMetricSpace
code: /-- Metric space structure on the completion of a pseudo_metric space. -/
instance instMetricSpace : MetricSpace (Completion α) 
goal: MetricSpace (UniformSpace.Completion α)

full_name: UniformSpace.Completion.eq_of_dist_eq_zero
code: @[deprecated eq_of_dist_eq_zero (since := "2024-03-10")]
protected theorem eq_of_dist_eq_zero (x y : Completion α) (h : dist x y = 0) : x = y 
goal: x = y

full_name: UniformSpace.Completion.coe_isometry
code: /-- The embedding of a metric space in its completion is an isometry. -/
theorem coe_isometry : Isometry ((↑) : α → Completion α) 
goal: Isometry ↑α

full_name: UniformSpace.Completion.edist_eq
code: @[simp]
protected theorem edist_eq (x y : α) : edist (x : Completion α) y = edist x y 
goal: edist (↑α x) (↑α y) = edist x y

full_name: LipschitzWith.completion_extension
code: theorem LipschitzWith.completion_extension [MetricSpace β] [CompleteSpace β] {f : α → β}
    {K : ℝ≥0} (h : LipschitzWith K f) : LipschitzWith K (Completion.extension f) 
goal: LipschitzWith K (UniformSpace.Completion.extension f)

full_name: LipschitzWith.completion_map
code: theorem LipschitzWith.completion_map [PseudoMetricSpace β] {f : α → β} {K : ℝ≥0}
    (h : LipschitzWith K f) : LipschitzWith K (Completion.map f) 
goal: LipschitzWith K (UniformSpace.Completion.map f)

full_name: Isometry.completion_extension
code: theorem Isometry.completion_extension [MetricSpace β] [CompleteSpace β] {f : α → β}
    (h : Isometry f) : Isometry (Completion.extension f) 
goal: Isometry (UniformSpace.Completion.extension f)

full_name: Isometry.completion_map
code: theorem Isometry.completion_map [PseudoMetricSpace β] {f : α → β}
    (h : Isometry f) : Isometry (Completion.map f) 
goal: Isometry (UniformSpace.Completion.map f)