File: Mathlib/Analysis/NormedSpace/Real.lean

full_name: Real.punctured_nhds_module_neBot
code: /-- If `E` is a nontrivial topological module over `ℝ`, then `E` has no isolated points.
This is a particular case of `Module.punctured_nhds_neBot`. -/
instance Real.punctured_nhds_module_neBot {E : Type*} [AddCommGroup E] [TopologicalSpace E]
    [ContinuousAdd E] [Nontrivial E] [Module ℝ E] [ContinuousSMul ℝ E] (x : E) : NeBot (𝓝[≠] x) 
goal: (𝓝[≠] x).NeBot

full_name: inv_norm_smul_mem_closed_unit_ball
code: theorem inv_norm_smul_mem_closed_unit_ball (x : E) :
    ‖x‖⁻¹ • x ∈ closedBall (0 : E) 1 
goal: ‖x‖⁻¹ • x ∈ Metric.closedBall 0 1

full_name: norm_smul_of_nonneg
code: theorem norm_smul_of_nonneg {t : ℝ} (ht : 0 ≤ t) (x : E) : ‖t • x‖ = t * ‖x‖ 
goal: ‖t • x‖ = t * ‖x‖

full_name: dist_smul_add_one_sub_smul_le
code: theorem dist_smul_add_one_sub_smul_le {r : ℝ} {x y : E} (h : r ∈ Icc 0 1) :
    dist (r • x + (1 - r) • y) x ≤ dist y x 
goal: dist (r • x + (1 - r) • y) x ≤ dist y x

full_name: closure_ball
code: theorem closure_ball (x : E) {r : ℝ} (hr : r ≠ 0) : closure (ball x r) = closedBall x r 
goal: closure (Metric.ball x r) = Metric.closedBall x r

full_name: frontier_ball
code: theorem frontier_ball (x : E) {r : ℝ} (hr : r ≠ 0) :
    frontier (ball x r) = sphere x r 
goal: frontier (Metric.ball x r) = Metric.sphere x r

full_name: interior_closedBall
code: theorem interior_closedBall (x : E) {r : ℝ} (hr : r ≠ 0) :
    interior (closedBall x r) = ball x r 
goal: interior (Metric.closedBall x r) = Metric.ball x r

full_name: frontier_closedBall
code: theorem frontier_closedBall (x : E) {r : ℝ} (hr : r ≠ 0) :
    frontier (closedBall x r) = sphere x r 
goal: frontier (Metric.closedBall x r) = Metric.sphere x r

full_name: interior_sphere
code: theorem interior_sphere (x : E) {r : ℝ} (hr : r ≠ 0) : interior (sphere x r) = ∅ 
goal: interior (Metric.sphere x r) = ∅

full_name: frontier_sphere
code: theorem frontier_sphere (x : E) {r : ℝ} (hr : r ≠ 0) : frontier (sphere x r) = sphere x r 
goal: frontier (Metric.sphere x r) = Metric.sphere x r

full_name: exists_norm_eq
code: theorem exists_norm_eq {c : ℝ} (hc : 0 ≤ c) : ∃ x : E, ‖x‖ = c 
goal: ∃ x, ‖x‖ = c

full_name: range_norm
code: @[simp]
theorem range_norm : range (norm : E → ℝ) = Ici 0 
goal: Set.range norm = Set.Ici 0

full_name: nnnorm_surjective
code: theorem nnnorm_surjective : Surjective (nnnorm : E → ℝ≥0) 
goal: Function.Surjective nnnorm

full_name: range_nnnorm
code: @[simp]
theorem range_nnnorm : range (nnnorm : E → ℝ≥0) = univ 
goal: Set.range nnnorm = Set.univ

full_name: interior_closedBall'
code: theorem interior_closedBall' (x : E) (r : ℝ) : interior (closedBall x r) = ball x r 
goal: interior (Metric.closedBall x r) = Metric.ball x r

full_name: frontier_closedBall'
code: theorem frontier_closedBall' (x : E) (r : ℝ) : frontier (closedBall x r) = sphere x r 
goal: frontier (Metric.closedBall x r) = Metric.sphere x r

full_name: interior_sphere'
code: @[simp]
theorem interior_sphere' (x : E) (r : ℝ) : interior (sphere x r) = ∅ 
goal: interior (Metric.sphere x r) = ∅

full_name: frontier_sphere'
code: @[simp]
theorem frontier_sphere' (x : E) (r : ℝ) : frontier (sphere x r) = sphere x r 
goal: frontier (Metric.sphere x r) = Metric.sphere x r