fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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continuous_sup [Max L] [ContinuousSup L] : Continuous fun p : L × L => p.1 ⊔ p.2 :=
ContinuousSup.continuous_sup
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | continuous_sup | null |
Continuous.sup [Max L] [ContinuousSup L] {f g : X → L} (hf : Continuous f)
(hg : Continuous g) : Continuous fun x => f x ⊔ g x :=
continuous_sup.comp (hf.prodMk hg :) | theorem | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.sup | null |
sup_nhds' [Max L] [ContinuousSup L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
Tendsto (f ⊔ g) l (𝓝 (x ⊔ y)) :=
(continuous_sup.tendsto _).comp (hf.prodMk_nhds hg) | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | sup_nhds' | null |
sup_nhds [Max 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)) :=
hf.sup_nhds' hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | sup_nhds | null |
inf_nhds' [Min L] [ContinuousInf L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
Tendsto (f ⊓ g) l (𝓝 (x ⊓ y)) :=
(continuous_inf.tendsto _).comp (hf.prodMk_nhds hg) | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | inf_nhds' | null |
inf_nhds [Min 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)) :=
hf.inf_nhds' hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | inf_nhds | null |
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)) := by
induction hne using Finset.Nonempty.cons_induction with
| singleton => simpa using hs
| cons a s ha hne ihs =>
rw [forall_mem_cons] at hs
simp only [sup'_cons, hne]
exact hs.1.sup_nhds (ihs hs.2) | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | finset_sup'_nhds | null |
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)) := by
simpa only [← Finset.sup'_apply] using finset_sup'_nhds hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | finset_sup'_nhds_apply | null |
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)) :=
finset_sup'_nhds (L := Lᵒᵈ) hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | finset_inf'_nhds | null |
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)) :=
finset_sup'_nhds_apply (L := Lᵒᵈ) hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | finset_inf'_nhds_apply | null |
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)) := by
rcases s.eq_empty_or_nonempty with rfl | hne
· simpa using tendsto_const_nhds
· simp only [← sup'_eq_sup hne]
exact finset_sup'_nhds hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | finset_sup_nhds | null |
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)) := by
simpa only [← Finset.sup_apply] using finset_sup_nhds hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | finset_sup_nhds_apply | null |
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)) :=
finset_sup_nhds (L := Lᵒᵈ) hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | finset_inf_nhds | null |
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)) :=
finset_sup_nhds_apply (L := Lᵒᵈ) hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | finset_inf_nhds_apply | null |
ContinuousAt.sup' (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (f ⊔ g) x :=
hf.sup_nhds' hg
@[fun_prop] | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.sup' | null |
ContinuousAt.sup (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (fun a ↦ f a ⊔ g a) x :=
hf.sup' hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.sup | null |
ContinuousWithinAt.sup' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (f ⊔ g) s x :=
hf.sup_nhds' hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.sup' | null |
ContinuousWithinAt.sup (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (fun a ↦ f a ⊔ g a) s x :=
hf.sup' hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.sup | null |
ContinuousOn.sup' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (f ⊔ g) s := fun x hx ↦
(hf x hx).sup' (hg x hx)
@[fun_prop] | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.sup' | null |
ContinuousOn.sup (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun a ↦ f a ⊔ g a) s :=
hf.sup' hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.sup | null |
Continuous.sup' (hf : Continuous f) (hg : Continuous g) : Continuous (f ⊔ g) := hf.sup hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.sup' | null |
ContinuousAt.inf' (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (f ⊓ g) x :=
hf.inf_nhds' hg
@[fun_prop] | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.inf' | null |
ContinuousAt.inf (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (fun a ↦ f a ⊓ g a) x :=
hf.inf' hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.inf | null |
ContinuousWithinAt.inf' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (f ⊓ g) s x :=
hf.inf_nhds' hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.inf' | null |
ContinuousWithinAt.inf (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (fun a ↦ f a ⊓ g a) s x :=
hf.inf' hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.inf | null |
ContinuousOn.inf' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (f ⊓ g) s := fun x hx ↦
(hf x hx).inf' (hg x hx)
@[fun_prop] | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.inf' | null |
ContinuousOn.inf (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun a ↦ f a ⊓ g a) s :=
hf.inf' hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.inf | null |
Continuous.inf' (hf : Continuous f) (hg : Continuous g) : Continuous (f ⊓ g) := hf.inf hg | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.inf' | null |
ContinuousAt.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (fun a ↦ s.sup' hne (f · a)) x :=
Tendsto.finset_sup'_nhds_apply hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.finset_sup'_apply | null |
ContinuousAt.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (s.sup' hne f) x := by
simpa only [← Finset.sup'_apply] using finset_sup'_apply hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.finset_sup' | null |
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 :=
Tendsto.finset_sup'_nhds_apply hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.finset_sup'_apply | null |
ContinuousWithinAt.finset_sup' (hne : s.Nonempty)
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.sup' hne f) t x := by
simpa only [← Finset.sup'_apply] using finset_sup'_apply hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.finset_sup' | null |
ContinuousOn.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (fun a ↦ s.sup' hne (f · a)) t := fun x hx ↦
ContinuousWithinAt.finset_sup'_apply hne fun i hi ↦ hs i hi x hx | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.finset_sup'_apply | null |
ContinuousOn.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (s.sup' hne f) t := fun x hx ↦
ContinuousWithinAt.finset_sup' hne fun i hi ↦ hs i hi x hx | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.finset_sup' | null |
Continuous.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (fun a ↦ s.sup' hne (f · a)) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup'_apply _ fun i hi ↦
(hs i hi).continuousAt | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.finset_sup'_apply | null |
Continuous.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (s.sup' hne f) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup' _ fun i hi ↦ (hs i hi).continuousAt | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.finset_sup' | null |
ContinuousAt.finset_sup_apply (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (fun a ↦ s.sup (f · a)) x :=
Tendsto.finset_sup_nhds_apply hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.finset_sup_apply | null |
ContinuousAt.finset_sup (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (s.sup f) x := by
simpa only [← Finset.sup_apply] using finset_sup_apply hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.finset_sup | null |
ContinuousWithinAt.finset_sup_apply
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
ContinuousWithinAt (fun a ↦ s.sup (f · a)) t x :=
Tendsto.finset_sup_nhds_apply hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.finset_sup_apply | null |
ContinuousWithinAt.finset_sup
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.sup f) t x := by
simpa only [← Finset.sup_apply] using finset_sup_apply hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.finset_sup | null |
ContinuousOn.finset_sup_apply (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (fun a ↦ s.sup (f · a)) t := fun x hx ↦
ContinuousWithinAt.finset_sup_apply fun i hi ↦ hs i hi x hx | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.finset_sup_apply | null |
ContinuousOn.finset_sup (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (s.sup f) t := fun x hx ↦
ContinuousWithinAt.finset_sup fun i hi ↦ hs i hi x hx | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.finset_sup | null |
Continuous.finset_sup_apply (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (fun a ↦ s.sup (f · a)) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup_apply fun i hi ↦
(hs i hi).continuousAt | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.finset_sup_apply | null |
Continuous.finset_sup (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.sup f) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup fun i hi ↦ (hs i hi).continuousAt | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.finset_sup | null |
ContinuousAt.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (fun a ↦ s.inf' hne (f · a)) x :=
Tendsto.finset_inf'_nhds_apply hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.finset_inf'_apply | null |
ContinuousAt.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (s.inf' hne f) x := by
simpa only [← Finset.inf'_apply] using finset_inf'_apply hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.finset_inf' | null |
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 :=
Tendsto.finset_inf'_nhds_apply hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.finset_inf'_apply | null |
ContinuousWithinAt.finset_inf' (hne : s.Nonempty)
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.inf' hne f) t x := by
simpa only [← Finset.inf'_apply] using finset_inf'_apply hne hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.finset_inf' | null |
ContinuousOn.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (fun a ↦ s.inf' hne (f · a)) t := fun x hx ↦
ContinuousWithinAt.finset_inf'_apply hne fun i hi ↦ hs i hi x hx | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.finset_inf'_apply | null |
ContinuousOn.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (s.inf' hne f) t := fun x hx ↦
ContinuousWithinAt.finset_inf' hne fun i hi ↦ hs i hi x hx | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.finset_inf' | null |
Continuous.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (fun a ↦ s.inf' hne (f · a)) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf'_apply _ fun i hi ↦
(hs i hi).continuousAt | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.finset_inf'_apply | null |
Continuous.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (s.inf' hne f) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf' _ fun i hi ↦ (hs i hi).continuousAt | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.finset_inf' | null |
ContinuousAt.finset_inf_apply (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (fun a ↦ s.inf (f · a)) x :=
Tendsto.finset_inf_nhds_apply hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.finset_inf_apply | null |
ContinuousAt.finset_inf (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (s.inf f) x := by
simpa only [← Finset.inf_apply] using finset_inf_apply hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousAt.finset_inf | null |
ContinuousWithinAt.finset_inf_apply
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) :
ContinuousWithinAt (fun a ↦ s.inf (f · a)) t x :=
Tendsto.finset_inf_nhds_apply hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.finset_inf_apply | null |
ContinuousWithinAt.finset_inf
(hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.inf f) t x := by
simpa only [← Finset.inf_apply] using finset_inf_apply hs | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousWithinAt.finset_inf | null |
ContinuousOn.finset_inf_apply (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (fun a ↦ s.inf (f · a)) t := fun x hx ↦
ContinuousWithinAt.finset_inf_apply fun i hi ↦ hs i hi x hx | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.finset_inf_apply | null |
ContinuousOn.finset_inf (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (s.inf f) t := fun x hx ↦
ContinuousWithinAt.finset_inf fun i hi ↦ hs i hi x hx | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousOn.finset_inf | null |
Continuous.finset_inf_apply (hs : ∀ i ∈ s, Continuous (f i)) :
Continuous (fun a ↦ s.inf (f · a)) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf_apply fun i hi ↦
(hs i hi).continuousAt | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.finset_inf_apply | null |
Continuous.finset_inf (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.inf f) :=
continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf fun i hi ↦ (hs i hi).continuousAt | lemma | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.finset_inf | null |
lawson (α : Type*) [Preorder α] : TopologicalSpace α := lower α ⊓ scott α univ
variable (α) [Preorder α] [TopologicalSpace α] | def | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | lawson | The Lawson topology is defined as the meet of `Topology.lower` and the `Topology.scott`. |
IsLawson : Prop where
topology_eq_lawson : ‹TopologicalSpace α› = lawson α | class | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | IsLawson | Predicate for an ordered topological space to be equipped with its Lawson topology.
The Lawson topology is defined as the meet of `Topology.lower` and the `Topology.scott`. |
lawsonBasis := { s : Set α | ∃ t : Set α, t.Finite ∧ ∃ u : Set α, IsOpen[scott α univ] u ∧
u \ upperClosure t = s } | def | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | lawsonBasis | The complements of the upper closures of finite sets intersected with Scott open sets form
a basis for the lawson topology. |
protected isTopologicalBasis : TopologicalSpace.IsTopologicalBasis (lawsonBasis α) := by
have lawsonBasis_image2 : lawsonBasis α =
(image2 (fun x x_1 ↦ ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1)
(IsLower.lowerBasis (WithLower α)) {U | IsOpen[scott α univ] U}) := by
rw [lawsonBasis, image2, IsLower.lowerBasis]
simp_rw [diff_eq_compl_inter]
aesop
rw [lawsonBasis_image2]
convert IsTopologicalBasis.inf_induced IsLower.isTopologicalBasis
(isTopologicalBasis_opens (α := WithScott α))
WithLower.toLower WithScott.toScott
rw [@topology_eq_lawson α _ _ _, lawson]
apply (congrArg₂ min _) _
· letI _ := lower α
exact (@IsLower.withLowerHomeomorph α ‹_› (lower α) ⟨rfl⟩).isInducing.eq_induced
· letI _ := scott α univ
exact (@IsScott.withScottHomeomorph α _ (scott α univ) ⟨rfl⟩).isInducing.eq_induced | theorem | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | isTopologicalBasis | null |
WithLawson (α : Type*) := α | def | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | WithLawson | Type synonym for a preorder equipped with the Lawson topology. |
@[match_pattern] toLawson : α ≃ WithLawson α := Equiv.refl _ | def | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | toLawson | `toLawson` is the identity function to the `WithLawson` of a type. |
@[match_pattern] ofLawson : WithLawson α ≃ α := Equiv.refl _
@[simp] lemma to_Lawson_symm_eq : (@toLawson α).symm = ofLawson := rfl
@[simp] lemma of_Lawson_symm_eq : (@ofLawson α).symm = toLawson := rfl
@[simp] lemma toLawson_ofLawson (a : WithLawson α) : toLawson (ofLawson a) = a := rfl
@[simp] lemma ofLawson_toLawson (a : α) : ofLawson (toLawson a) = a := rfl | def | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | ofLawson | `ofLawson` is the identity function from the `WithLawson` of a type. |
toLawson_inj {a b : α} : toLawson a = toLawson b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | toLawson_inj | null |
ofLawson_inj {a b : WithLawson α} : ofLawson a = ofLawson b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | ofLawson_inj | null |
@[elab_as_elim, cases_eliminator, induction_eliminator]
protected rec {β : WithLawson α → Sort*}
(h : ∀ a, β (toLawson a)) : ∀ a, β a := fun a => h (ofLawson a) | def | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | rec | A recursor for `WithLawson`. Use as `induction x`. |
instPreorder : Preorder (WithLawson α) := ‹Preorder α› | instance | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | instPreorder | null |
instTopologicalSpace : TopologicalSpace (WithLawson α) := lawson α | instance | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | instTopologicalSpace | null |
instIsLawson : IsLawson (WithLawson α) := ⟨rfl⟩ | instance | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | instIsLawson | null |
homeomorph [TopologicalSpace α] [IsLawson α] : WithLawson α ≃ₜ α :=
ofLawson.toHomeomorphOfIsInducing ⟨IsLawson.topology_eq_lawson (α := α) ▸ induced_id.symm⟩ | def | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | homeomorph | If `α` is equipped with the Lawson topology, then it is homeomorphic to `WithLawson α`. |
isOpen_preimage_ofLawson {S : Set α} :
IsOpen (ofLawson ⁻¹' S) ↔ (lawson α).IsOpen S := Iff.rfl | theorem | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | isOpen_preimage_ofLawson | null |
isClosed_preimage_ofLawson {S : Set α} :
IsClosed (ofLawson ⁻¹' S) ↔ IsClosed[lawson α] S := Iff.rfl | theorem | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | isClosed_preimage_ofLawson | null |
isOpen_def {T : Set (WithLawson α)} :
IsOpen T ↔ (lawson α).IsOpen (toLawson ⁻¹' T) := Iff.rfl | theorem | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | isOpen_def | null |
lawson_le_scott : lawson α ≤ scott α univ := inf_le_right | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | lawson_le_scott | null |
lawson_le_lower : lawson α ≤ lower α := inf_le_left | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | lawson_le_lower | null |
scottHausdorff_le_lawson : scottHausdorff α univ ≤ lawson α :=
le_inf scottHausdorff_le_lower scottHausdorff_le_scott | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | scottHausdorff_le_lawson | null |
lawsonClosed_of_scottClosed (s : Set α) (h : IsClosed (WithScott.ofScott ⁻¹' s)) :
IsClosed (WithLawson.ofLawson ⁻¹' s) := h.mono lawson_le_scott | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | lawsonClosed_of_scottClosed | null |
lawsonClosed_of_lowerClosed (s : Set α) (h : IsClosed (WithLower.ofLower ⁻¹' s)) :
IsClosed (WithLawson.ofLawson ⁻¹' s) := h.mono lawson_le_lower | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | lawsonClosed_of_lowerClosed | null |
lawsonOpen_iff_scottOpen_of_isUpperSet {s : Set α} (h : IsUpperSet s) :
IsOpen (WithLawson.ofLawson ⁻¹' s) ↔ IsOpen (WithScott.ofScott ⁻¹' s) :=
⟨fun hs => IsScott.isOpen_iff_isUpperSet_and_scottHausdorff_open (D := univ).mpr
⟨h, (scottHausdorff_le_lawson s) hs⟩, lawson_le_scott _⟩
variable (L : TopologicalSpace α) (S : TopologicalSpace α)
variable [@IsLawson α _ L] [@IsScott α univ _ S] | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | lawsonOpen_iff_scottOpen_of_isUpperSet | An upper set is Lawson open if and only if it is Scott open |
isLawson_le_isScott : L ≤ S := by
rw [@IsScott.topology_eq α univ _ S _, @IsLawson.topology_eq_lawson α _ L _]
exact inf_le_right | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | isLawson_le_isScott | null |
scottHausdorff_le_isLawson : scottHausdorff α univ ≤ L := by
rw [@IsLawson.topology_eq_lawson α _ L _]
exact scottHausdorff_le_lawson | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | scottHausdorff_le_isLawson | null |
lawsonOpen_iff_scottOpen_of_isUpperSet' (s : Set α) (h : IsUpperSet s) :
IsOpen[L] s ↔ IsOpen[S] s := by
rw [@IsLawson.topology_eq_lawson α _ L _, @IsScott.topology_eq α univ _ S _]
exact lawsonOpen_iff_scottOpen_of_isUpperSet h | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | lawsonOpen_iff_scottOpen_of_isUpperSet' | An upper set is Lawson open if and only if it is Scott open |
lawsonClosed_iff_scottClosed_of_isLowerSet (s : Set α) (h : IsLowerSet s) :
IsClosed[L] s ↔ IsClosed[S] s := by
rw [← @isOpen_compl_iff, ← isOpen_compl_iff,
(lawsonOpen_iff_scottOpen_of_isUpperSet' L S _ (isUpperSet_compl.mpr h))]
include S in | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | lawsonClosed_iff_scottClosed_of_isLowerSet | null |
lawsonClosed_iff_dirSupClosed_of_isLowerSet (s : Set α) (h : IsLowerSet s) :
IsClosed[L] s ↔ DirSupClosed s := by
rw [lawsonClosed_iff_scottClosed_of_isLowerSet L S _ h,
@IsScott.isClosed_iff_isLowerSet_and_dirSupClosed]
simp_all | lemma | Topology | [
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Order.ScottTopology"
] | Mathlib/Topology/Order/LawsonTopology.lean | lawsonClosed_iff_dirSupClosed_of_isLowerSet | A lower set is Lawson closed if and only if it is closed under sups of directed sets |
frequently_lt_nhds (a : α) [NeBot (𝓝[<] a)] : ∃ᶠ x in 𝓝 a, x < a :=
frequently_iff_neBot.2 ‹_› | lemma | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | frequently_lt_nhds | null |
frequently_gt_nhds (a : α) [NeBot (𝓝[>] a)] : ∃ᶠ x in 𝓝 a, a < x :=
frequently_iff_neBot.2 ‹_› | lemma | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | frequently_gt_nhds | null |
Filter.Eventually.exists_lt {a : α} [NeBot (𝓝[<] a)] {p : α → Prop}
(h : ∀ᶠ x in 𝓝 a, p x) : ∃ b < a, p b :=
((frequently_lt_nhds a).and_eventually h).exists | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | Filter.Eventually.exists_lt | null |
Filter.Eventually.exists_gt {a : α} [NeBot (𝓝[>] a)] {p : α → Prop}
(h : ∀ᶠ x in 𝓝 a, p x) : ∃ b > a, p b :=
((frequently_gt_nhds a).and_eventually h).exists | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | Filter.Eventually.exists_gt | null |
nhdsWithin_Ici_neBot {a b : α} (H₂ : a ≤ b) : NeBot (𝓝[Ici a] b) :=
nhdsWithin_neBot_of_mem H₂ | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsWithin_Ici_neBot | null |
nhdsGE_neBot (a : α) : NeBot (𝓝[≥] a) := nhdsWithin_Ici_neBot (le_refl a) | instance | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsGE_neBot | null |
nhdsWithin_Iic_neBot {a b : α} (H : a ≤ b) : NeBot (𝓝[Iic b] a) :=
nhdsWithin_neBot_of_mem H | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsWithin_Iic_neBot | null |
nhdsLE_neBot (a : α) : NeBot (𝓝[≤] a) := nhdsWithin_Iic_neBot (le_refl a) | instance | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsLE_neBot | null |
nhdsLT_le_nhdsNE (a : α) : 𝓝[<] a ≤ 𝓝[≠] a :=
nhdsWithin_mono a fun _ => ne_of_lt | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsLT_le_nhdsNE | null |
nhdsGT_le_nhdsNE (a : α) : 𝓝[>] a ≤ 𝓝[≠] a := nhdsWithin_mono a fun _ => ne_of_gt | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsGT_le_nhdsNE | null |
IsAntichain.interior_eq_empty [∀ x : α, (𝓝[<] x).NeBot] {s : Set α}
(hs : IsAntichain (· ≤ ·) s) : interior s = ∅ := by
refine eq_empty_of_forall_notMem fun x hx ↦ ?_
have : ∀ᶠ y in 𝓝 x, y ∈ s := mem_interior_iff_mem_nhds.1 hx
rcases this.exists_lt with ⟨y, hyx, hys⟩
exact hs hys (interior_subset hx) hyx.ne hyx.le | lemma | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | IsAntichain.interior_eq_empty | null |
IsAntichain.interior_eq_empty' [∀ x : α, (𝓝[>] x).NeBot] {s : Set α}
(hs : IsAntichain (· ≤ ·) s) : interior s = ∅ :=
have : ∀ x : αᵒᵈ, NeBot (𝓝[<] x) := ‹_›
hs.to_dual.interior_eq_empty | lemma | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | IsAntichain.interior_eq_empty' | null |
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