File: Mathlib/Analysis/LocallyConvex/Basic.lean

full_name: Balanced
code: /-- A set `A` is balanced if `a • A` is contained in `A` whenever `a` has norm at most `1`. -/
def Balanced (A : Set E) 
goal: Prop

full_name: absorbs_iff_norm
code: lemma absorbs_iff_norm : Absorbs 𝕜 A B ↔ ∃ r, ∀ c : 𝕜, r ≤ ‖c‖ → B ⊆ c • A 
goal: Absorbs 𝕜 A B ↔ ∃ r, ∀ (c : 𝕜), r ≤ ‖c‖ → B ⊆ c • A

full_name: Absorbs.exists_pos
code: lemma Absorbs.exists_pos (h : Absorbs 𝕜 A B) : ∃ r > 0, ∀ c : 𝕜, r ≤ ‖c‖ → B ⊆ c • A 
goal: ∃ r > 0, ∀ (c : 𝕜), r ≤ ‖c‖ → B ⊆ c • A

full_name: balanced_iff_smul_mem
code: theorem balanced_iff_smul_mem : Balanced 𝕜 s ↔ ∀ ⦃a : 𝕜⦄, ‖a‖ ≤ 1 → ∀ ⦃x : E⦄, x ∈ s → a • x ∈ s 
goal: Balanced 𝕜 s ↔ ∀ ⦃a : 𝕜⦄, ‖a‖ ≤ 1 → ∀ ⦃x : E⦄, x ∈ s → a • x ∈ s

full_name: balanced_iff_closedBall_smul
code: theorem balanced_iff_closedBall_smul : Balanced 𝕜 s ↔ Metric.closedBall (0 : 𝕜) 1 • s ⊆ s 
goal: Balanced 𝕜 s ↔ Metric.closedBall 0 1 • s ⊆ s

full_name: balanced_empty
code: @[simp]
theorem balanced_empty : Balanced 𝕜 (∅ : Set E) 
goal: Balanced 𝕜 ∅

full_name: balanced_univ
code: @[simp]
theorem balanced_univ : Balanced 𝕜 (univ : Set E) 
goal: Balanced 𝕜 Set.univ

full_name: Balanced.union
code: theorem Balanced.union (hA : Balanced 𝕜 A) (hB : Balanced 𝕜 B) : Balanced 𝕜 (A ∪ B) 
goal: Balanced 𝕜 (A ∪ B)

full_name: Balanced.inter
code: theorem Balanced.inter (hA : Balanced 𝕜 A) (hB : Balanced 𝕜 B) : Balanced 𝕜 (A ∩ B) 
goal: Balanced 𝕜 (A ∩ B)

full_name: balanced_iUnion
code: theorem balanced_iUnion {f : ι → Set E} (h : ∀ i, Balanced 𝕜 (f i)) : Balanced 𝕜 (⋃ i, f i) 
goal: Balanced 𝕜 (⋃ i, f i)

full_name: balanced_iUnion₂
code: theorem balanced_iUnion₂ {f : ∀ i, κ i → Set E} (h : ∀ i j, Balanced 𝕜 (f i j)) :
    Balanced 𝕜 (⋃ (i) (j), f i j) 
goal: Balanced 𝕜 (⋃ i, ⋃ j, f i j)

full_name: balanced_iInter
code: theorem balanced_iInter {f : ι → Set E} (h : ∀ i, Balanced 𝕜 (f i)) : Balanced 𝕜 (⋂ i, f i) 
goal: Balanced 𝕜 (⋂ i, f i)

full_name: balanced_iInter₂
code: theorem balanced_iInter₂ {f : ∀ i, κ i → Set E} (h : ∀ i j, Balanced 𝕜 (f i j)) :
    Balanced 𝕜 (⋂ (i) (j), f i j) 
goal: Balanced 𝕜 (⋂ i, ⋂ j, f i j)

full_name: Balanced.smul
code: theorem Balanced.smul (a : 𝕝) (hs : Balanced 𝕜 s) : Balanced 𝕜 (a • s) 
goal: Balanced 𝕜 (a • s)

full_name: Balanced.neg
code: theorem Balanced.neg : Balanced 𝕜 s → Balanced 𝕜 (-s) 
goal: Balanced 𝕜 s → Balanced 𝕜 (-s)

full_name: balanced_neg
code: @[simp]
theorem balanced_neg : Balanced 𝕜 (-s) ↔ Balanced 𝕜 s 
goal: Balanced 𝕜 (-s) ↔ Balanced 𝕜 s

full_name: Balanced.neg_mem_iff
code: theorem Balanced.neg_mem_iff [NormOneClass 𝕜] (h : Balanced 𝕜 s) {x : E} : -x ∈ s ↔ x ∈ s 
goal: -x ∈ s ↔ x ∈ s

full_name: Balanced.neg_eq
code: theorem Balanced.neg_eq [NormOneClass 𝕜] (h : Balanced 𝕜 s) : -s = s 
goal: -s = s

full_name: Balanced.add
code: theorem Balanced.add (hs : Balanced 𝕜 s) (ht : Balanced 𝕜 t) : Balanced 𝕜 (s + t) 
goal: Balanced 𝕜 (s + t)

full_name: Balanced.sub
code: theorem Balanced.sub (hs : Balanced 𝕜 s) (ht : Balanced 𝕜 t) : Balanced 𝕜 (s - t) 
goal: Balanced 𝕜 (s - t)

full_name: balanced_zero
code: theorem balanced_zero : Balanced 𝕜 (0 : Set E) 
goal: Balanced 𝕜 0

full_name: absorbs_iff_eventually_nhdsWithin_zero
code: theorem absorbs_iff_eventually_nhdsWithin_zero :
    Absorbs 𝕜 s t ↔ ∀ᶠ c : 𝕜 in 𝓝[≠] 0, MapsTo (c • ·) t s 
goal: Absorbs 𝕜 s t ↔ ∀ᶠ (c : 𝕜) in 𝓝[≠] 0, Set.MapsTo (fun x => c • x) t s

full_name: absorbent_iff_eventually_nhdsWithin_zero
code: theorem absorbent_iff_eventually_nhdsWithin_zero :
    Absorbent 𝕜 s ↔ ∀ x : E, ∀ᶠ c : 𝕜 in 𝓝[≠] 0, c • x ∈ s 
goal: Absorbent 𝕜 s ↔ ∀ (x : E), ∀ᶠ (c : 𝕜) in 𝓝[≠] 0, c • x ∈ s

full_name: absorbs_iff_eventually_nhds_zero
code: theorem absorbs_iff_eventually_nhds_zero (h₀ : 0 ∈ s) :
    Absorbs 𝕜 s t ↔ ∀ᶠ c : 𝕜 in 𝓝 0, MapsTo (c • ·) t s 
goal: Absorbs 𝕜 s t ↔ ∀ᶠ (c : 𝕜) in 𝓝 0, Set.MapsTo (fun x => c • x) t s

full_name: Absorbs.eventually_nhds_zero
code: theorem Absorbs.eventually_nhds_zero (h : Absorbs 𝕜 s t) (h₀ : 0 ∈ s) :
    ∀ᶠ c : 𝕜 in 𝓝 0, MapsTo (c • ·) t s 
goal: ∀ᶠ (c : 𝕜) in 𝓝 0, Set.MapsTo (fun x => c • x) t s

full_name: Balanced.smul_mono
code: /-- Scalar multiplication (by possibly different types) of a balanced set is monotone. -/
theorem Balanced.smul_mono (hs : Balanced 𝕝 s) {a : 𝕝} {b : 𝕜} (h : ‖a‖ ≤ ‖b‖) : a • s ⊆ b • s 
goal: a • s ⊆ b • s

full_name: Balanced.smul_mem_mono
code: theorem Balanced.smul_mem_mono [SMulCommClass 𝕝 𝕜 E] (hs : Balanced 𝕝 s) {a : 𝕜} {b : 𝕝}
    (ha : a • x ∈ s) (hba : ‖b‖ ≤ ‖a‖) : b • x ∈ s 
goal: b • x ∈ s

full_name: Balanced.subset_smul
code: theorem Balanced.subset_smul (hA : Balanced 𝕜 A) (ha : 1 ≤ ‖a‖) : A ⊆ a • A 
goal: A ⊆ a • A

full_name: Balanced.smul_congr
code: theorem Balanced.smul_congr (hs : Balanced 𝕜 A) (h : ‖a‖ = ‖b‖) : a • A = b • A 
goal: a • A = b • A

full_name: Balanced.smul_eq
code: theorem Balanced.smul_eq (hA : Balanced 𝕜 A) (ha : ‖a‖ = 1) : a • A = A 
goal: a • A = A

full_name: Balanced.absorbs_self
code: /-- A balanced set absorbs itself. -/
theorem Balanced.absorbs_self (hA : Balanced 𝕜 A) : Absorbs 𝕜 A A 
goal: Absorbs 𝕜 A A

full_name: Balanced.smul_mem_iff
code: theorem Balanced.smul_mem_iff (hs : Balanced 𝕜 s) (h : ‖a‖ = ‖b‖) : a • x ∈ s ↔ b • x ∈ s 
goal: a • x ∈ s ↔ b • x ∈ s

full_name: absorbent_nhds_zero
code: /-- Every neighbourhood of the origin is absorbent. -/
theorem absorbent_nhds_zero (hA : A ∈ 𝓝 (0 : E)) : Absorbent 𝕜 A 
goal: Absorbent 𝕜 A

full_name: Balanced.zero_insert_interior
code: /-- The union of `{0}` with the interior of a balanced set is balanced. -/
theorem Balanced.zero_insert_interior (hA : Balanced 𝕜 A) :
    Balanced 𝕜 (insert 0 (interior A)) 
goal: Balanced 𝕜 (insert 0 (interior A))

full_name: balanced_zero_union_interior
code: @[deprecated Balanced.zero_insert_interior (since := "2024-02-03")]
theorem balanced_zero_union_interior (hA : Balanced 𝕜 A) : Balanced 𝕜 ((0 : Set E) ∪ interior A) 
goal: Balanced 𝕜 (0 ∪ interior A)

full_name: Balanced.interior
code: /-- The interior of a balanced set is balanced if it contains the origin. -/
protected theorem Balanced.interior (hA : Balanced 𝕜 A) (h : (0 : E) ∈ interior A) :
    Balanced 𝕜 (interior A) 
goal: Balanced 𝕜 (interior A)

full_name: Balanced.closure
code: protected theorem Balanced.closure (hA : Balanced 𝕜 A) : Balanced 𝕜 (closure A) 
goal: Balanced 𝕜 (closure A)

full_name: Absorbent.zero_mem'
code: @[deprecated Absorbent.zero_mem (since := "2024-02-02")]
theorem Absorbent.zero_mem' (hs : Absorbent 𝕜 s) : (0 : E) ∈ s 
goal: 0 ∈ s

full_name: Balanced.convexHull
code: protected theorem Balanced.convexHull (hs : Balanced 𝕜 s) : Balanced 𝕜 (convexHull ℝ s) 
goal: Balanced 𝕜 ((convexHull ℝ) s)

full_name: balanced_iff_neg_mem
code: theorem balanced_iff_neg_mem (hs : Convex ℝ s) : Balanced ℝ s ↔ ∀ ⦃x⦄, x ∈ s → -x ∈ s 
goal: Balanced ℝ s ↔ ∀ ⦃x : E⦄, x ∈ s → -x ∈ s