File: Mathlib/Analysis/NormedSpace/Pointwise.lean

full_name: ediam_smul_le
code: theorem ediam_smul_le (c : 𝕜) (s : Set E) : EMetric.diam (c • s) ≤ ‖c‖₊ • EMetric.diam s 
goal: EMetric.diam (c • s) ≤ ‖c‖₊ • EMetric.diam s

full_name: ediam_smul₀
code: theorem ediam_smul₀ (c : 𝕜) (s : Set E) : EMetric.diam (c • s) = ‖c‖₊ • EMetric.diam s 
goal: EMetric.diam (c • s) = ‖c‖₊ • EMetric.diam s

full_name: diam_smul₀
code: theorem diam_smul₀ (c : 𝕜) (x : Set E) : diam (c • x) = ‖c‖ * diam x 
goal: Metric.diam (c • x) = ‖c‖ * Metric.diam x

full_name: infEdist_smul₀
code: theorem infEdist_smul₀ {c : 𝕜} (hc : c ≠ 0) (s : Set E) (x : E) :
    EMetric.infEdist (c • x) (c • s) = ‖c‖₊ • EMetric.infEdist x s 
goal: EMetric.infEdist (c • x) (c • s) = ‖c‖₊ • EMetric.infEdist x s

full_name: infDist_smul₀
code: theorem infDist_smul₀ {c : 𝕜} (hc : c ≠ 0) (s : Set E) (x : E) :
    Metric.infDist (c • x) (c • s) = ‖c‖ * Metric.infDist x s 
goal: Metric.infDist (c • x) (c • s) = ‖c‖ * Metric.infDist x s

full_name: smul_ball
code: theorem smul_ball {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) : c • ball x r = ball (c • x) (‖c‖ * r) 
goal: c • Metric.ball x r = Metric.ball (c • x) (‖c‖ * r)

full_name: smul_unitBall
code: theorem smul_unitBall {c : 𝕜} (hc : c ≠ 0) : c • ball (0 : E) (1 : ℝ) = ball (0 : E) ‖c‖ 
goal: c • Metric.ball 0 1 = Metric.ball 0 ‖c‖

full_name: smul_sphere'
code: theorem smul_sphere' {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) :
    c • sphere x r = sphere (c • x) (‖c‖ * r) 
goal: c • Metric.sphere x r = Metric.sphere (c • x) (‖c‖ * r)

full_name: smul_closedBall'
code: theorem smul_closedBall' {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) :
    c • closedBall x r = closedBall (c • x) (‖c‖ * r) 
goal: c • Metric.closedBall x r = Metric.closedBall (c • x) (‖c‖ * r)

full_name: set_smul_sphere_zero
code: theorem set_smul_sphere_zero {s : Set 𝕜} (hs : 0 ∉ s) (r : ℝ) :
    s • sphere (0 : E) r = (‖·‖) ⁻¹' ((‖·‖ * r) '' s) 
goal: s • Metric.sphere 0 r = (fun x => ‖x‖) ⁻¹' ((fun x => ‖x‖ * r) '' s)

full_name: Bornology.IsBounded.smul₀
code: /-- Image of a bounded set in a normed space under scalar multiplication by a constant is
bounded. See also `Bornology.IsBounded.smul` for a similar lemma about an isometric action. -/
theorem Bornology.IsBounded.smul₀ {s : Set E} (hs : IsBounded s) (c : 𝕜) : IsBounded (c • s) 
goal: Bornology.IsBounded (c • s)

full_name: eventually_singleton_add_smul_subset
code: /-- If `s` is a bounded set, then for small enough `r`, the set `{x} + r • s` is contained in any
fixed neighborhood of `x`. -/
theorem eventually_singleton_add_smul_subset {x : E} {s : Set E} (hs : Bornology.IsBounded s)
    {u : Set E} (hu : u ∈ 𝓝 x) : ∀ᶠ r in 𝓝 (0 : 𝕜), {x} + r • s ⊆ u 
goal: ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u

full_name: smul_unitBall_of_pos
code: /-- In a real normed space, the image of the unit ball under scalar multiplication by a positive
constant `r` is the ball of radius `r`. -/
theorem smul_unitBall_of_pos {r : ℝ} (hr : 0 < r) : r • ball (0 : E) 1 = ball (0 : E) r 
goal: r • Metric.ball 0 1 = Metric.ball 0 r

full_name: Ioo_smul_sphere_zero
code: lemma Ioo_smul_sphere_zero {a b r : ℝ} (ha : 0 ≤ a) (hr : 0 < r) :
    Ioo a b • sphere (0 : E) r = ball 0 (b * r) \ closedBall 0 (a * r) 
goal: Set.Ioo a b • Metric.sphere 0 r = Metric.ball 0 (b * r) \ Metric.closedBall 0 (a * r)

full_name: exists_dist_eq
code: theorem exists_dist_eq (x z : E) {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
    ∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z 
goal: ∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z

full_name: exists_dist_le_le
code: theorem exists_dist_le_le (hδ : 0 ≤ δ) (hε : 0 ≤ ε) (h : dist x z ≤ ε + δ) :
    ∃ y, dist x y ≤ δ ∧ dist y z ≤ ε 
goal: ∃ y, dist x y ≤ δ ∧ dist y z ≤ ε

full_name: exists_dist_le_lt
code: theorem exists_dist_le_lt (hδ : 0 ≤ δ) (hε : 0 < ε) (h : dist x z < ε + δ) :
    ∃ y, dist x y ≤ δ ∧ dist y z < ε 
goal: ∃ y, dist x y ≤ δ ∧ dist y z < ε

full_name: exists_dist_lt_le
code: theorem exists_dist_lt_le (hδ : 0 < δ) (hε : 0 ≤ ε) (h : dist x z < ε + δ) :
    ∃ y, dist x y < δ ∧ dist y z ≤ ε 
goal: ∃ y, dist x y < δ ∧ dist y z ≤ ε

full_name: exists_dist_lt_lt
code: theorem exists_dist_lt_lt (hδ : 0 < δ) (hε : 0 < ε) (h : dist x z < ε + δ) :
    ∃ y, dist x y < δ ∧ dist y z < ε 
goal: ∃ y, dist x y < δ ∧ dist y z < ε

full_name: disjoint_ball_ball_iff
code: theorem disjoint_ball_ball_iff (hδ : 0 < δ) (hε : 0 < ε) :
    Disjoint (ball x δ) (ball y ε) ↔ δ + ε ≤ dist x y 
goal: Disjoint (Metric.ball x δ) (Metric.ball y ε) ↔ δ + ε ≤ dist x y

full_name: disjoint_ball_closedBall_iff
code: theorem disjoint_ball_closedBall_iff (hδ : 0 < δ) (hε : 0 ≤ ε) :
    Disjoint (ball x δ) (closedBall y ε) ↔ δ + ε ≤ dist x y 
goal: Disjoint (Metric.ball x δ) (Metric.closedBall y ε) ↔ δ + ε ≤ dist x y

full_name: disjoint_closedBall_ball_iff
code: theorem disjoint_closedBall_ball_iff (hδ : 0 ≤ δ) (hε : 0 < ε) :
    Disjoint (closedBall x δ) (ball y ε) ↔ δ + ε ≤ dist x y 
goal: Disjoint (Metric.closedBall x δ) (Metric.ball y ε) ↔ δ + ε ≤ dist x y

full_name: disjoint_closedBall_closedBall_iff
code: theorem disjoint_closedBall_closedBall_iff (hδ : 0 ≤ δ) (hε : 0 ≤ ε) :
    Disjoint (closedBall x δ) (closedBall y ε) ↔ δ + ε < dist x y 
goal: Disjoint (Metric.closedBall x δ) (Metric.closedBall y ε) ↔ δ + ε < dist x y

full_name: infEdist_thickening
code: @[simp]
theorem infEdist_thickening (hδ : 0 < δ) (s : Set E) (x : E) :
    infEdist x (thickening δ s) = infEdist x s - ENNReal.ofReal δ 
goal: EMetric.infEdist x (Metric.thickening δ s) = EMetric.infEdist x s - ENNReal.ofReal δ

full_name: thickening_thickening
code: @[simp]
theorem thickening_thickening (hε : 0 < ε) (hδ : 0 < δ) (s : Set E) :
    thickening ε (thickening δ s) = thickening (ε + δ) s 
goal: Metric.thickening ε (Metric.thickening δ s) = Metric.thickening (ε + δ) s

full_name: cthickening_thickening
code: @[simp]
theorem cthickening_thickening (hε : 0 ≤ ε) (hδ : 0 < δ) (s : Set E) :
    cthickening ε (thickening δ s) = cthickening (ε + δ) s 
goal: Metric.cthickening ε (Metric.thickening δ s) = Metric.cthickening (ε + δ) s

full_name: closure_thickening
code: @[simp]
theorem closure_thickening (hδ : 0 < δ) (s : Set E) :
    closure (thickening δ s) = cthickening δ s 
goal: closure (Metric.thickening δ s) = Metric.cthickening δ s

full_name: infEdist_cthickening
code: @[simp]
theorem infEdist_cthickening (δ : ℝ) (s : Set E) (x : E) :
    infEdist x (cthickening δ s) = infEdist x s - ENNReal.ofReal δ 
goal: EMetric.infEdist x (Metric.cthickening δ s) = EMetric.infEdist x s - ENNReal.ofReal δ

full_name: thickening_cthickening
code: @[simp]
theorem thickening_cthickening (hε : 0 < ε) (hδ : 0 ≤ δ) (s : Set E) :
    thickening ε (cthickening δ s) = thickening (ε + δ) s 
goal: Metric.thickening ε (Metric.cthickening δ s) = Metric.thickening (ε + δ) s

full_name: cthickening_cthickening
code: @[simp]
theorem cthickening_cthickening (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (s : Set E) :
    cthickening ε (cthickening δ s) = cthickening (ε + δ) s 
goal: Metric.cthickening ε (Metric.cthickening δ s) = Metric.cthickening (ε + δ) s

full_name: thickening_ball
code: @[simp]
theorem thickening_ball (hε : 0 < ε) (hδ : 0 < δ) (x : E) :
    thickening ε (ball x δ) = ball x (ε + δ) 
goal: Metric.thickening ε (Metric.ball x δ) = Metric.ball x (ε + δ)

full_name: thickening_closedBall
code: @[simp]
theorem thickening_closedBall (hε : 0 < ε) (hδ : 0 ≤ δ) (x : E) :
    thickening ε (closedBall x δ) = ball x (ε + δ) 
goal: Metric.thickening ε (Metric.closedBall x δ) = Metric.ball x (ε + δ)

full_name: cthickening_ball
code: @[simp]
theorem cthickening_ball (hε : 0 ≤ ε) (hδ : 0 < δ) (x : E) :
    cthickening ε (ball x δ) = closedBall x (ε + δ) 
goal: Metric.cthickening ε (Metric.ball x δ) = Metric.closedBall x (ε + δ)

full_name: cthickening_closedBall
code: @[simp]
theorem cthickening_closedBall (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (x : E) :
    cthickening ε (closedBall x δ) = closedBall x (ε + δ) 
goal: Metric.cthickening ε (Metric.closedBall x δ) = Metric.closedBall x (ε + δ)

full_name: ball_add_ball
code: theorem ball_add_ball (hε : 0 < ε) (hδ : 0 < δ) (a b : E) :
    ball a ε + ball b δ = ball (a + b) (ε + δ) 
goal: Metric.ball a ε + Metric.ball b δ = Metric.ball (a + b) (ε + δ)

full_name: ball_sub_ball
code: theorem ball_sub_ball (hε : 0 < ε) (hδ : 0 < δ) (a b : E) :
    ball a ε - ball b δ = ball (a - b) (ε + δ) 
goal: Metric.ball a ε - Metric.ball b δ = Metric.ball (a - b) (ε + δ)

full_name: ball_add_closedBall
code: theorem ball_add_closedBall (hε : 0 < ε) (hδ : 0 ≤ δ) (a b : E) :
    ball a ε + closedBall b δ = ball (a + b) (ε + δ) 
goal: Metric.ball a ε + Metric.closedBall b δ = Metric.ball (a + b) (ε + δ)

full_name: ball_sub_closedBall
code: theorem ball_sub_closedBall (hε : 0 < ε) (hδ : 0 ≤ δ) (a b : E) :
    ball a ε - closedBall b δ = ball (a - b) (ε + δ) 
goal: Metric.ball a ε - Metric.closedBall b δ = Metric.ball (a - b) (ε + δ)

full_name: closedBall_add_ball
code: theorem closedBall_add_ball (hε : 0 ≤ ε) (hδ : 0 < δ) (a b : E) :
    closedBall a ε + ball b δ = ball (a + b) (ε + δ) 
goal: Metric.closedBall a ε + Metric.ball b δ = Metric.ball (a + b) (ε + δ)

full_name: closedBall_sub_ball
code: theorem closedBall_sub_ball (hε : 0 ≤ ε) (hδ : 0 < δ) (a b : E) :
    closedBall a ε - ball b δ = ball (a - b) (ε + δ) 
goal: Metric.closedBall a ε - Metric.ball b δ = Metric.ball (a - b) (ε + δ)

full_name: closedBall_add_closedBall
code: theorem closedBall_add_closedBall [ProperSpace E] (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (a b : E) :
    closedBall a ε + closedBall b δ = closedBall (a + b) (ε + δ) 
goal: Metric.closedBall a ε + Metric.closedBall b δ = Metric.closedBall (a + b) (ε + δ)

full_name: closedBall_sub_closedBall
code: theorem closedBall_sub_closedBall [ProperSpace E] (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (a b : E) :
    closedBall a ε - closedBall b δ = closedBall (a - b) (ε + δ) 
goal: Metric.closedBall a ε - Metric.closedBall b δ = Metric.closedBall (a - b) (ε + δ)

full_name: smul_closedBall
code: theorem smul_closedBall (c : 𝕜) (x : E) {r : ℝ} (hr : 0 ≤ r) :
    c • closedBall x r = closedBall (c • x) (‖c‖ * r) 
goal: c • Metric.closedBall x r = Metric.closedBall (c • x) (‖c‖ * r)

full_name: smul_closedUnitBall
code: theorem smul_closedUnitBall (c : 𝕜) : c • closedBall (0 : E) (1 : ℝ) = closedBall (0 : E) ‖c‖ 
goal: c • Metric.closedBall 0 1 = Metric.closedBall 0 ‖c‖

full_name: smul_closedUnitBall_of_nonneg
code: /-- In a real normed space, the image of the unit closed ball under multiplication by a nonnegative
number `r` is the closed ball of radius `r` with center at the origin. -/
theorem smul_closedUnitBall_of_nonneg {r : ℝ} (hr : 0 ≤ r) :
    r • closedBall (0 : E) 1 = closedBall (0 : E) r 
goal: r • Metric.closedBall 0 1 = Metric.closedBall 0 r

full_name: NormedSpace.sphere_nonempty
code: /-- In a nontrivial real normed space, a sphere is nonempty if and only if its radius is
nonnegative. -/
@[simp]
theorem NormedSpace.sphere_nonempty [Nontrivial E] {x : E} {r : ℝ} :
    (sphere x r).Nonempty ↔ 0 ≤ r 
goal: (Metric.sphere x r).Nonempty ↔ 0 ≤ r

full_name: smul_sphere
code: theorem smul_sphere [Nontrivial E] (c : 𝕜) (x : E) {r : ℝ} (hr : 0 ≤ r) :
    c • sphere x r = sphere (c • x) (‖c‖ * r) 
goal: c • Metric.sphere x r = Metric.sphere (c • x) (‖c‖ * r)

full_name: affinity_unitBall
code: /-- Any ball `Metric.ball x r`, `0 < r` is the image of the unit ball under `fun y ↦ x + r • y`. -/
theorem affinity_unitBall {r : ℝ} (hr : 0 < r) (x : E) : x +ᵥ r • ball (0 : E) 1 = ball x r 
goal: x +ᵥ r • Metric.ball 0 1 = Metric.ball x r

full_name: affinity_unitClosedBall
code: /-- Any closed ball `Metric.closedBall x r`, `0 ≤ r` is the image of the unit closed ball under
`fun y ↦ x + r • y`. -/
theorem affinity_unitClosedBall {r : ℝ} (hr : 0 ≤ r) (x : E) :
    x +ᵥ r • closedBall (0 : E) 1 = closedBall x r 
goal: x +ᵥ r • Metric.closedBall 0 1 = Metric.closedBall x r