File: Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean

full_name: Real.dist_left_le_of_mem_uIcc
code: lemma Real.dist_left_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist x y ≤ dist x z 
goal: dist x y ≤ dist x z

full_name: Real.dist_right_le_of_mem_uIcc
code: lemma Real.dist_right_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist y z ≤ dist x z 
goal: dist y z ≤ dist x z

full_name: Real.dist_le_of_mem_uIcc
code: lemma Real.dist_le_of_mem_uIcc {x y x' y' : ℝ} (hx : x ∈ uIcc x' y') (hy : y ∈ uIcc x' y') :
    dist x y ≤ dist x' y' 
goal: dist x y ≤ dist x' y'

full_name: Real.dist_le_of_mem_Icc
code: lemma Real.dist_le_of_mem_Icc {x y x' y' : ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
    dist x y ≤ y' - x' 
goal: dist x y ≤ y' - x'

full_name: Real.dist_le_of_mem_Icc_01
code: lemma Real.dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1) :
    dist x y ≤ 1 
goal: dist x y ≤ 1

full_name: instOrderTopologyReal
code: instance : OrderTopology ℝ 
goal: OrderTopology ℝ

full_name: Real.singleton_eq_inter_Icc
code: lemma Real.singleton_eq_inter_Icc (b : ℝ) : {b} = ⋂ (r > 0), Icc (b - r) (b + r) 
goal: {b} = ⋂ r, ⋂ (_ : r > 0), Set.Icc (b - r) (b + r)

full_name: squeeze_zero'
code: /-- Special case of the sandwich lemma; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the
general case. -/
lemma squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t in t₀, 0 ≤ f t)
    (hft : ∀ᶠ t in t₀, f t ≤ g t) (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) 
goal: Filter.Tendsto f t₀ (𝓝 0)

full_name: squeeze_zero
code: /-- Special case of the sandwich lemma; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
and `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/
lemma squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 ≤ f t) (hft : ∀ t, f t ≤ g t)
    (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) 
goal: Filter.Tendsto f t₀ (𝓝 0)

full_name: eventually_closedBall_subset
code: /-- If `u` is a neighborhood of `x`, then for small enough `r`, the closed ball
`Metric.closedBall x r` is contained in `u`. -/
lemma eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
    ∀ᶠ r in 𝓝 (0 : ℝ), closedBall x r ⊆ u 
goal: ∀ᶠ (r : ℝ) in 𝓝 0, Metric.closedBall x r ⊆ u

full_name: tendsto_closedBall_smallSets
code: lemma tendsto_closedBall_smallSets (x : α) : Tendsto (closedBall x) (𝓝 0) (𝓝 x).smallSets 
goal: Filter.Tendsto (Metric.closedBall x) (𝓝 0) (𝓝 x).smallSets

full_name: Metric.isClosed_ball
code: lemma isClosed_ball : IsClosed (closedBall x ε) 
goal: IsClosed (Metric.closedBall x ε)

full_name: Metric.isClosed_sphere
code: lemma isClosed_sphere : IsClosed (sphere x ε) 
goal: IsClosed (Metric.sphere x ε)

full_name: Metric.closure_closedBall
code: @[simp]
lemma closure_closedBall : closure (closedBall x ε) = closedBall x ε 
goal: closure (Metric.closedBall x ε) = Metric.closedBall x ε

full_name: Metric.closure_sphere
code: @[simp]
lemma closure_sphere : closure (sphere x ε) = sphere x ε 
goal: closure (Metric.sphere x ε) = Metric.sphere x ε

full_name: Metric.closure_ball_subset_closedBall
code: lemma closure_ball_subset_closedBall : closure (ball x ε) ⊆ closedBall x ε 
goal: closure (Metric.ball x ε) ⊆ Metric.closedBall x ε

full_name: Metric.frontier_ball_subset_sphere
code: lemma frontier_ball_subset_sphere : frontier (ball x ε) ⊆ sphere x ε 
goal: frontier (Metric.ball x ε) ⊆ Metric.sphere x ε

full_name: Metric.frontier_closedBall_subset_sphere
code: lemma frontier_closedBall_subset_sphere : frontier (closedBall x ε) ⊆ sphere x ε 
goal: frontier (Metric.closedBall x ε) ⊆ Metric.sphere x ε

full_name: Metric.closedBall_zero'
code: lemma closedBall_zero' (x : α) : closedBall x 0 = closure {x} 
goal: Metric.closedBall x 0 = closure {x}

full_name: Metric.eventually_isCompact_closedBall
code: lemma eventually_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) :
    ∀ᶠ r in 𝓝 (0 : ℝ), IsCompact (closedBall x r) 
goal: ∀ᶠ (r : ℝ) in 𝓝 0, IsCompact (Metric.closedBall x r)

full_name: Metric.exists_isCompact_closedBall
code: lemma exists_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) :
    ∃ r, 0 < r ∧ IsCompact (closedBall x r) 
goal: ∃ r, 0 < r ∧ IsCompact (Metric.closedBall x r)

full_name: Real.dist_le_of_mem_pi_Icc
code: lemma dist_le_of_mem_pi_Icc (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') : dist x y ≤ dist x' y' 
goal: dist x y ≤ dist x' y'