File: Mathlib/Topology/Bornology/Constructions.lean

full_name: Prod.instBornology
code: instance Prod.instBornology : Bornology (α × β) 
goal: Bornology (α × β)

full_name: Pi.instBornology
code: instance Pi.instBornology : Bornology (∀ i, π i) 
goal: Bornology ((i : ι) → π i)

full_name: Bornology.induced
code: /-- Inverse image of a bornology. -/
abbrev Bornology.induced {α β : Type*} [Bornology β] (f : α → β) : Bornology α 
goal: Bornology α

full_name: instBornologySubtype
code: instance {p : α → Prop} : Bornology (Subtype p) 
goal: Bornology (Subtype p)

full_name: Bornology.cobounded_prod
code: theorem cobounded_prod : cobounded (α × β) = (cobounded α).coprod (cobounded β) 
goal: Bornology.cobounded (α × β) = (Bornology.cobounded α).coprod (Bornology.cobounded β)

full_name: Bornology.isBounded_image_fst_and_snd
code: theorem isBounded_image_fst_and_snd {s : Set (α × β)} :
    IsBounded (Prod.fst '' s) ∧ IsBounded (Prod.snd '' s) ↔ IsBounded s 
goal: Bornology.IsBounded (Prod.fst '' s) ∧ Bornology.IsBounded (Prod.snd '' s) ↔ Bornology.IsBounded s

full_name: Bornology.IsBounded.image_fst
code: lemma IsBounded.image_fst {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.fst '' s) 
goal: Bornology.IsBounded (Prod.fst '' s)

full_name: Bornology.IsBounded.image_snd
code: lemma IsBounded.image_snd {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.snd '' s) 
goal: Bornology.IsBounded (Prod.snd '' s)

full_name: Bornology.IsBounded.fst_of_prod
code: theorem IsBounded.fst_of_prod (h : IsBounded (s ×ˢ t)) (ht : t.Nonempty) : IsBounded s 
goal: Bornology.IsBounded s

full_name: Bornology.IsBounded.snd_of_prod
code: theorem IsBounded.snd_of_prod (h : IsBounded (s ×ˢ t)) (hs : s.Nonempty) : IsBounded t 
goal: Bornology.IsBounded t

full_name: Bornology.IsBounded.prod
code: theorem IsBounded.prod (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ×ˢ t) 
goal: Bornology.IsBounded (s ×ˢ t)

full_name: Bornology.isBounded_prod_of_nonempty
code: theorem isBounded_prod_of_nonempty (hne : Set.Nonempty (s ×ˢ t)) :
    IsBounded (s ×ˢ t) ↔ IsBounded s ∧ IsBounded t 
goal: Bornology.IsBounded (s ×ˢ t) ↔ Bornology.IsBounded s ∧ Bornology.IsBounded t

full_name: Bornology.isBounded_prod
code: theorem isBounded_prod : IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ IsBounded s ∧ IsBounded t 
goal: Bornology.IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ Bornology.IsBounded s ∧ Bornology.IsBounded t

full_name: Bornology.isBounded_prod_self
code: theorem isBounded_prod_self : IsBounded (s ×ˢ s) ↔ IsBounded s 
goal: Bornology.IsBounded (s ×ˢ s) ↔ Bornology.IsBounded s

full_name: Bornology.cobounded_pi
code: theorem cobounded_pi : cobounded (∀ i, π i) = Filter.coprodᵢ fun i => cobounded (π i) 
goal: Bornology.cobounded ((i : ι) → π i) = Filter.coprodᵢ fun i => Bornology.cobounded (π i)

full_name: Bornology.forall_isBounded_image_eval_iff
code: theorem forall_isBounded_image_eval_iff {s : Set (∀ i, π i)} :
    (∀ i, IsBounded (eval i '' s)) ↔ IsBounded s 
goal: (∀ (i : ι), Bornology.IsBounded (Function.eval i '' s)) ↔ Bornology.IsBounded s

full_name: Bornology.IsBounded.image_eval
code: lemma IsBounded.image_eval {s : Set (∀ i, π i)} (hs : IsBounded s) (i : ι) :
    IsBounded (eval i '' s) 
goal: Bornology.IsBounded (Function.eval i '' s)

full_name: Bornology.IsBounded.pi
code: theorem IsBounded.pi (h : ∀ i, IsBounded (S i)) : IsBounded (pi univ S) 
goal: Bornology.IsBounded (Set.univ.pi S)

full_name: Bornology.isBounded_pi_of_nonempty
code: theorem isBounded_pi_of_nonempty (hne : (pi univ S).Nonempty) :
    IsBounded (pi univ S) ↔ ∀ i, IsBounded (S i) 
goal: Bornology.IsBounded (Set.univ.pi S) ↔ ∀ (i : ι), Bornology.IsBounded (S i)

full_name: Bornology.isBounded_pi
code: theorem isBounded_pi : IsBounded (pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ i, IsBounded (S i) 
goal: Bornology.IsBounded (Set.univ.pi S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), Bornology.IsBounded (S i)

full_name: Bornology.isBounded_induced
code: theorem isBounded_induced {α β : Type*} [Bornology β] {f : α → β} {s : Set α} :
    @IsBounded α (Bornology.induced f) s ↔ IsBounded (f '' s) 
goal: Bornology.IsBounded s ↔ Bornology.IsBounded (f '' s)

full_name: Bornology.isBounded_image_subtype_val
code: theorem isBounded_image_subtype_val {p : α → Prop} {s : Set { x // p x }} :
    IsBounded (Subtype.val '' s) ↔ IsBounded s 
goal: Bornology.IsBounded (Subtype.val '' s) ↔ Bornology.IsBounded s

full_name: instBoundedSpaceProd
code: instance [BoundedSpace α] [BoundedSpace β] : BoundedSpace (α × β) 
goal: BoundedSpace (α × β)

full_name: instBoundedSpaceForall
code: instance [∀ i, BoundedSpace (π i)] : BoundedSpace (∀ i, π i) 
goal: BoundedSpace ((i : ι) → π i)

full_name: boundedSpace_induced_iff
code: theorem boundedSpace_induced_iff {α β : Type*} [Bornology β] {f : α → β} :
    @BoundedSpace α (Bornology.induced f) ↔ IsBounded (range f) 
goal: BoundedSpace α ↔ Bornology.IsBounded (Set.range f)

full_name: boundedSpace_subtype_iff
code: theorem boundedSpace_subtype_iff {p : α → Prop} :
    BoundedSpace (Subtype p) ↔ IsBounded { x | p x } 
goal: BoundedSpace (Subtype p) ↔ Bornology.IsBounded {x | p x}

full_name: boundedSpace_val_set_iff
code: theorem boundedSpace_val_set_iff {s : Set α} : BoundedSpace s ↔ IsBounded s 
goal: BoundedSpace ↑s ↔ Bornology.IsBounded s

full_name: instBoundedSpaceSubtype
code: instance [BoundedSpace α] {p : α → Prop} : BoundedSpace (Subtype p) 
goal: BoundedSpace (Subtype p)

full_name: instBornologyAdditive
code: instance : Bornology (Additive α) 
goal: Bornology (Additive α)

full_name: instBornologyMultiplicative
code: instance : Bornology (Multiplicative α) 
goal: Bornology (Multiplicative α)

full_name: instBoundedSpaceAdditive
code: instance [BoundedSpace α] : BoundedSpace (Additive α) 
goal: BoundedSpace (Additive α)

full_name: instBoundedSpaceMultiplicative
code: instance [BoundedSpace α] : BoundedSpace (Multiplicative α) 
goal: BoundedSpace (Multiplicative α)

full_name: instBornologyOrderDual
code: instance : Bornology αᵒᵈ 
goal: Bornology αᵒᵈ

full_name: instBoundedSpaceOrderDual
code: instance [BoundedSpace α] : BoundedSpace αᵒᵈ 
goal: BoundedSpace αᵒᵈ