File: Mathlib/Topology/Order/Lattice.lean

full_name: ContinuousInf.continuous_inf
code: /-- Let `L` be a topological space and let `L×L` be equipped with the product topology and let
`⊓:L×L → L` be an infimum. Then `L` is said to have *(jointly) continuous infimum* if the map
`⊓:L×L → L` is continuous.
-/
class ContinuousInf (L : Type*) [TopologicalSpace L] [Inf L] : Prop where
  /-- The infimum is continuous -/
  continuous_inf : Continuous fun p : L × L => p.1 ⊓ p.2
goal: Continuous fun p => p.1 ⊓ p.2

full_name: ContinuousInf
code: /-- Let `L` be a topological space and let `L×L` be equipped with the product topology and let
`⊓:L×L → L` be an infimum. Then `L` is said to have *(jointly) continuous infimum* if the map
`⊓:L×L → L` is continuous.
-/
class ContinuousInf (L : Type*) [TopologicalSpace L] [Inf L] : Prop where
  /-- The infimum is continuous -/
  continuous_inf : Continuous fun p : L × L => p.1 ⊓ p.2
goal: Prop

full_name: ContinuousSup
code: /-- Let `L` be a topological space and let `L×L` be equipped with the product topology and let
`⊓:L×L → L` be a supremum. Then `L` is said to have *(jointly) continuous supremum* if the map
`⊓:L×L → L` is continuous.
-/
class ContinuousSup (L : Type*) [TopologicalSpace L] [Sup L] : Prop where
  /-- The supremum is continuous -/
  continuous_sup : Continuous fun p : L × L => p.1 ⊔ p.2
goal: Prop

full_name: ContinuousSup.continuous_sup
code: /-- Let `L` be a topological space and let `L×L` be equipped with the product topology and let
`⊓:L×L → L` be a supremum. Then `L` is said to have *(jointly) continuous supremum* if the map
`⊓:L×L → L` is continuous.
-/
class ContinuousSup (L : Type*) [TopologicalSpace L] [Sup L] : Prop where
  /-- The supremum is continuous -/
  continuous_sup : Continuous fun p : L × L => p.1 ⊔ p.2
goal: Continuous fun p => p.1 ⊔ p.2

full_name: OrderDual.continuousSup
code: instance (priority := 100) OrderDual.continuousSup (L : Type*) [TopologicalSpace L] [Inf L]
    [ContinuousInf L] : ContinuousSup Lᵒᵈ 
goal: ContinuousSup Lᵒᵈ

full_name: OrderDual.continuousInf
code: instance (priority := 100) OrderDual.continuousInf (L : Type*) [TopologicalSpace L] [Sup L]
    [ContinuousSup L] : ContinuousInf Lᵒᵈ 
goal: ContinuousInf Lᵒᵈ

full_name: TopologicalLattice
code: /-- Let `L` be a lattice equipped with a topology such that `L` has continuous infimum and supremum.
Then `L` is said to be a *topological lattice*.
-/
class TopologicalLattice (L : Type*) [TopologicalSpace L] [Lattice L]
  extends ContinuousInf L, ContinuousSup L : Prop
goal: Prop

full_name: OrderDual.topologicalLattice
code: instance (priority := 100) OrderDual.topologicalLattice (L : Type*) [TopologicalSpace L]
    [Lattice L] [TopologicalLattice L] : TopologicalLattice Lᵒᵈ 
goal: TopologicalLattice Lᵒᵈ

full_name: LinearOrder.topologicalLattice
code: instance (priority := 100) LinearOrder.topologicalLattice {L : Type*} [TopologicalSpace L]
    [LinearOrder L] [OrderClosedTopology L] : TopologicalLattice L 
goal: TopologicalLattice L

full_name: continuous_inf
code: @[continuity]
theorem continuous_inf [Inf L] [ContinuousInf L] : Continuous fun p : L × L => p.1 ⊓ p.2 
goal: Continuous fun p => p.1 ⊓ p.2

full_name: Continuous.inf
code: @[continuity, fun_prop]
theorem Continuous.inf [Inf L] [ContinuousInf L] {f g : X → L} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun x => f x ⊓ g x 
goal: Continuous fun x => f x ⊓ g x

full_name: continuous_sup
code: @[continuity]
theorem continuous_sup [Sup L] [ContinuousSup L] : Continuous fun p : L × L => p.1 ⊔ p.2 
goal: Continuous fun p => p.1 ⊔ p.2

full_name: Continuous.sup
code: @[continuity, fun_prop]
theorem Continuous.sup [Sup L] [ContinuousSup L] {f g : X → L} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun x => f x ⊔ g x 
goal: Continuous fun x => f x ⊔ g x

full_name: Filter.Tendsto.sup_nhds'
code: lemma sup_nhds' [Sup L] [ContinuousSup L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
    Tendsto (f ⊔ g) l (𝓝 (x ⊔ y)) 
goal: Filter.Tendsto (f ⊔ g) l (𝓝 (x ⊔ y))

full_name: Filter.Tendsto.sup_nhds
code: lemma sup_nhds [Sup L] [ContinuousSup L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
    Tendsto (fun i => f i ⊔ g i) l (𝓝 (x ⊔ y)) 
goal: Filter.Tendsto (fun i => f i ⊔ g i) l (𝓝 (x ⊔ y))

full_name: Filter.Tendsto.inf_nhds'
code: lemma inf_nhds' [Inf L] [ContinuousInf L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
    Tendsto (f ⊓ g) l (𝓝 (x ⊓ y)) 
goal: Filter.Tendsto (f ⊓ g) l (𝓝 (x ⊓ y))

full_name: Filter.Tendsto.inf_nhds
code: lemma inf_nhds [Inf L] [ContinuousInf L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
    Tendsto (fun i => f i ⊓ g i) l (𝓝 (x ⊓ y)) 
goal: Filter.Tendsto (fun i => f i ⊓ g i) l (𝓝 (x ⊓ y))

full_name: Filter.Tendsto.finset_sup'_nhds
code: lemma finset_sup'_nhds [SemilatticeSup L] [ContinuousSup L]
    (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (s.sup' hne f) l (𝓝 (s.sup' hne g)) 
goal: Filter.Tendsto (s.sup' hne f) l (𝓝 (s.sup' hne g))

full_name: Filter.Tendsto.finset_sup'_nhds_apply
code: lemma finset_sup'_nhds_apply [SemilatticeSup L] [ContinuousSup L]
    (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (fun a ↦ s.sup' hne (f · a)) l (𝓝 (s.sup' hne g)) 
goal: Filter.Tendsto (fun a => s.sup' hne fun x => f x a) l (𝓝 (s.sup' hne g))

full_name: Filter.Tendsto.finset_inf'_nhds
code: lemma finset_inf'_nhds [SemilatticeInf L] [ContinuousInf L]
    (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (s.inf' hne f) l (𝓝 (s.inf' hne g)) 
goal: Filter.Tendsto (s.inf' hne f) l (𝓝 (s.inf' hne g))

full_name: Filter.Tendsto.finset_inf'_nhds_apply
code: lemma finset_inf'_nhds_apply [SemilatticeInf L] [ContinuousInf L]
    (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (fun a ↦ s.inf' hne (f · a)) l (𝓝 (s.inf' hne g)) 
goal: Filter.Tendsto (fun a => s.inf' hne fun x => f x a) l (𝓝 (s.inf' hne g))

full_name: Filter.Tendsto.finset_sup_nhds
code: lemma finset_sup_nhds [SemilatticeSup L] [OrderBot L] [ContinuousSup L]
    (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (s.sup f) l (𝓝 (s.sup g)) 
goal: Filter.Tendsto (s.sup f) l (𝓝 (s.sup g))

full_name: Filter.Tendsto.finset_sup_nhds_apply
code: lemma finset_sup_nhds_apply [SemilatticeSup L] [OrderBot L] [ContinuousSup L]
    (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (fun a ↦ s.sup (f · a)) l (𝓝 (s.sup g)) 
goal: Filter.Tendsto (fun a => s.sup fun x => f x a) l (𝓝 (s.sup g))

full_name: Filter.Tendsto.finset_inf_nhds
code: lemma finset_inf_nhds [SemilatticeInf L] [OrderTop L] [ContinuousInf L]
    (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (s.inf f) l (𝓝 (s.inf g)) 
goal: Filter.Tendsto (s.inf f) l (𝓝 (s.inf g))

full_name: Filter.Tendsto.finset_inf_nhds_apply
code: lemma finset_inf_nhds_apply [SemilatticeInf L] [OrderTop L] [ContinuousInf L]
    (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
    Tendsto (fun a ↦ s.inf (f · a)) l (𝓝 (s.inf g)) 
goal: Filter.Tendsto (fun a => s.inf fun x => f x a) l (𝓝 (s.inf g))

full_name: ContinuousAt.sup'
code: lemma ContinuousAt.sup' (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (f ⊔ g) x 
goal: ContinuousAt (f ⊔ g) x

full_name: ContinuousAt.sup
code: @[fun_prop]
lemma ContinuousAt.sup (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (fun a ↦ f a ⊔ g a) x 
goal: ContinuousAt (fun a => f a ⊔ g a) x

full_name: ContinuousWithinAt.sup'
code: lemma ContinuousWithinAt.sup' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (f ⊔ g) s x 
goal: ContinuousWithinAt (f ⊔ g) s x

full_name: ContinuousWithinAt.sup
code: lemma ContinuousWithinAt.sup (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (fun a ↦ f a ⊔ g a) s x 
goal: ContinuousWithinAt (fun a => f a ⊔ g a) s x

full_name: ContinuousOn.sup'
code: lemma ContinuousOn.sup' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (f ⊔ g) s 
goal: ContinuousOn (f ⊔ g) s

full_name: ContinuousOn.sup
code: @[fun_prop]
lemma ContinuousOn.sup (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun a ↦ f a ⊔ g a) s 
goal: ContinuousOn (fun a => f a ⊔ g a) s

full_name: Continuous.sup'
code: lemma Continuous.sup' (hf : Continuous f) (hg : Continuous g) : Continuous (f ⊔ g) 
goal: Continuous (f ⊔ g)

full_name: ContinuousAt.inf'
code: lemma ContinuousAt.inf' (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (f ⊓ g) x 
goal: ContinuousAt (f ⊓ g) x

full_name: ContinuousAt.inf
code: @[fun_prop]
lemma ContinuousAt.inf (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
    ContinuousAt (fun a ↦ f a ⊓ g a) x 
goal: ContinuousAt (fun a => f a ⊓ g a) x

full_name: ContinuousWithinAt.inf'
code: lemma ContinuousWithinAt.inf' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (f ⊓ g) s x 
goal: ContinuousWithinAt (f ⊓ g) s x

full_name: ContinuousWithinAt.inf
code: lemma ContinuousWithinAt.inf (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (fun a ↦ f a ⊓ g a) s x 
goal: ContinuousWithinAt (fun a => f a ⊓ g a) s x

full_name: ContinuousOn.inf'
code: lemma ContinuousOn.inf' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (f ⊓ g) s 
goal: ContinuousOn (f ⊓ g) s

full_name: ContinuousOn.inf
code: @[fun_prop]
lemma ContinuousOn.inf (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun a ↦ f a ⊓ g a) s 
goal: ContinuousOn (fun a => f a ⊓ g a) s

full_name: Continuous.inf'
code: lemma Continuous.inf' (hf : Continuous f) (hg : Continuous g) : Continuous (f ⊓ g) 
goal: Continuous (f ⊓ g)

full_name: ContinuousAt.finset_sup'_apply
code: lemma ContinuousAt.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (fun a ↦ s.sup' hne (f · a)) x 
goal: ContinuousAt (fun a => s.sup' hne fun x => f x a) x

full_name: ContinuousAt.finset_sup'
code: lemma ContinuousAt.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (s.sup' hne f) x 
goal: ContinuousAt (s.sup' hne f) x

full_name: ContinuousWithinAt.finset_sup'_apply
code: lemma ContinuousWithinAt.finset_sup'_apply (hne : s.Nonempty)
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
    ContinuousWithinAt (fun a ↦ s.sup' hne (f · a)) t x 
goal: ContinuousWithinAt (fun a => s.sup' hne fun x => f x a) t x

full_name: ContinuousWithinAt.finset_sup'
code: lemma ContinuousWithinAt.finset_sup' (hne : s.Nonempty)
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.sup' hne f) t x 
goal: ContinuousWithinAt (s.sup' hne f) t x

full_name: ContinuousOn.finset_sup'_apply
code: lemma ContinuousOn.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (fun a ↦ s.sup' hne (f · a)) t 
goal: ContinuousOn (fun a => s.sup' hne fun x => f x a) t

full_name: ContinuousOn.finset_sup'
code: lemma ContinuousOn.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (s.sup' hne f) t 
goal: ContinuousOn (s.sup' hne f) t

full_name: Continuous.finset_sup'_apply
code: lemma Continuous.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (fun a ↦ s.sup' hne (f · a)) 
goal: Continuous fun a => s.sup' hne fun x => f x a

full_name: Continuous.finset_sup'
code: lemma Continuous.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (s.sup' hne f) 
goal: Continuous (s.sup' hne f)

full_name: ContinuousAt.finset_sup_apply
code: lemma ContinuousAt.finset_sup_apply (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (fun a ↦ s.sup (f · a)) x 
goal: ContinuousAt (fun a => s.sup fun x => f x a) x

full_name: ContinuousAt.finset_sup
code: lemma ContinuousAt.finset_sup (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (s.sup f) x 
goal: ContinuousAt (s.sup f) x

full_name: ContinuousWithinAt.finset_sup_apply
code: lemma ContinuousWithinAt.finset_sup_apply
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
    ContinuousWithinAt (fun a ↦ s.sup (f · a)) t x 
goal: ContinuousWithinAt (fun a => s.sup fun x => f x a) t x

full_name: ContinuousWithinAt.finset_sup
code: lemma ContinuousWithinAt.finset_sup
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.sup f) t x 
goal: ContinuousWithinAt (s.sup f) t x

full_name: ContinuousOn.finset_sup_apply
code: lemma ContinuousOn.finset_sup_apply (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (fun a ↦ s.sup (f · a)) t 
goal: ContinuousOn (fun a => s.sup fun x => f x a) t

full_name: ContinuousOn.finset_sup
code: lemma ContinuousOn.finset_sup (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (s.sup f) t 
goal: ContinuousOn (s.sup f) t

full_name: Continuous.finset_sup_apply
code: lemma Continuous.finset_sup_apply (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (fun a ↦ s.sup (f · a)) 
goal: Continuous fun a => s.sup fun x => f x a

full_name: Continuous.finset_sup
code: lemma Continuous.finset_sup (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.sup f) 
goal: Continuous (s.sup f)

full_name: ContinuousAt.finset_inf'_apply
code: lemma ContinuousAt.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (fun a ↦ s.inf' hne (f · a)) x 
goal: ContinuousAt (fun a => s.inf' hne fun x => f x a) x

full_name: ContinuousAt.finset_inf'
code: lemma ContinuousAt.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (s.inf' hne f) x 
goal: ContinuousAt (s.inf' hne f) x

full_name: ContinuousWithinAt.finset_inf'_apply
code: lemma ContinuousWithinAt.finset_inf'_apply (hne : s.Nonempty)
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
    ContinuousWithinAt (fun a ↦ s.inf' hne (f · a)) t x 
goal: ContinuousWithinAt (fun a => s.inf' hne fun x => f x a) t x

full_name: ContinuousWithinAt.finset_inf'
code: lemma ContinuousWithinAt.finset_inf' (hne : s.Nonempty)
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.inf' hne f) t x 
goal: ContinuousWithinAt (s.inf' hne f) t x

full_name: ContinuousOn.finset_inf'_apply
code: lemma ContinuousOn.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (fun a ↦ s.inf' hne (f · a)) t 
goal: ContinuousOn (fun a => s.inf' hne fun x => f x a) t

full_name: ContinuousOn.finset_inf'
code: lemma ContinuousOn.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (s.inf' hne f) t 
goal: ContinuousOn (s.inf' hne f) t

full_name: Continuous.finset_inf'_apply
code: lemma Continuous.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (fun a ↦ s.inf' hne (f · a)) 
goal: Continuous fun a => s.inf' hne fun x => f x a

full_name: Continuous.finset_inf'
code: lemma Continuous.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (s.inf' hne f) 
goal: Continuous (s.inf' hne f)

full_name: ContinuousAt.finset_inf_apply
code: lemma ContinuousAt.finset_inf_apply (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (fun a ↦ s.inf (f · a)) x 
goal: ContinuousAt (fun a => s.inf fun x => f x a) x

full_name: ContinuousAt.finset_inf
code: lemma ContinuousAt.finset_inf (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (s.inf f) x 
goal: ContinuousAt (s.inf f) x

full_name: ContinuousWithinAt.finset_inf_apply
code: lemma ContinuousWithinAt.finset_inf_apply
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
    ContinuousWithinAt (fun a ↦ s.inf (f · a)) t x 
goal: ContinuousWithinAt (fun a => s.inf fun x => f x a) t x

full_name: ContinuousWithinAt.finset_inf
code: lemma ContinuousWithinAt.finset_inf
    (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.inf f) t x 
goal: ContinuousWithinAt (s.inf f) t x

full_name: ContinuousOn.finset_inf_apply
code: lemma ContinuousOn.finset_inf_apply (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (fun a ↦ s.inf (f · a)) t 
goal: ContinuousOn (fun a => s.inf fun x => f x a) t

full_name: ContinuousOn.finset_inf
code: lemma ContinuousOn.finset_inf (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
    ContinuousOn (s.inf f) t 
goal: ContinuousOn (s.inf f) t

full_name: Continuous.finset_inf_apply
code: lemma Continuous.finset_inf_apply (hs : ∀ i ∈ s, Continuous (f i)) :
    Continuous (fun a ↦ s.inf (f · a)) 
goal: Continuous fun a => s.inf fun x => f x a

full_name: Continuous.finset_inf
code: lemma Continuous.finset_inf (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.inf f) 
goal: Continuous (s.inf f)