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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Filter.EventuallyEq.isLocalExtr_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) :
IsLocalExtr f a ↔ IsLocalExtr g a :=
heq.isExtrFilter_iff heq.eq_of_nhds | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | Filter.EventuallyEq.isLocalExtr_iff | null |
isLocalMax_of_mono_anti' {α : Type*} [TopologicalSpace α] [LinearOrder α]
{β : Type*} [Preorder β] {b : α} {f : α → β}
{a : Set α} (ha : a ∈ 𝓝[≤] b) {c : Set α} (hc : c ∈ 𝓝[≥] b)
(h₀ : MonotoneOn f a) (h₁ : AntitoneOn f c) : IsLocalMax f b :=
have : b ∈ a := mem_of_mem_nhdsWithin (by simp) ha
have : b ∈ c := mem_of_mem_nhdsWithin (by simp) hc
mem_of_superset (nhds_of_Ici_Iic ha hc) (fun x _ => by rcases le_total x b <;> aesop) | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalMax_of_mono_anti' | If `f` is monotone to the left and antitone to the right, then it has a local maximum. |
isLocalMin_of_anti_mono' {α : Type*} [TopologicalSpace α] [LinearOrder α]
{β : Type*} [Preorder β] {b : α} {f : α → β}
{a : Set α} (ha : a ∈ 𝓝[≤] b) {c : Set α} (hc : c ∈ 𝓝[≥] b)
(h₀ : AntitoneOn f a) (h₁ : MonotoneOn f c) : IsLocalMin f b :=
have : b ∈ a := mem_of_mem_nhdsWithin (by simp) ha
have : b ∈ c := mem_of_mem_nhdsWithin (by simp) hc
mem_of_superset (nhds_of_Ici_Iic ha hc) (fun x _ => by rcases le_total x b <;> aesop) | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LocalExtr.lean | isLocalMin_of_anti_mono' | If `f` is antitone to the left and monotone to the right, then it has a local minimum. |
lower (α : Type*) [Preorder α] : TopologicalSpace α := generateFrom {s | ∃ a, (Ici a)ᶜ = s} | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | lower | The lower topology is the topology generated by the complements of the left-closed right-infinite
intervals. |
upper (α : Type*) [Preorder α] : TopologicalSpace α := generateFrom {s | ∃ a, (Iic a)ᶜ = s} | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | upper | The upper topology is the topology generated by the complements of the right-closed left-infinite
intervals. |
WithLower (α : Type*) := α
variable {α β : Type*} | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | WithLower | Type synonym for a preorder equipped with the lower set topology. |
@[match_pattern] toLower : α ≃ WithLower α := Equiv.refl _ | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | toLower | `toLower` is the identity function to the `WithLower` of a type. |
@[match_pattern] ofLower : WithLower α ≃ α := Equiv.refl _
@[simp] lemma toLower_symm : (@toLower α).symm = ofLower := rfl
@[simp] lemma ofLower_symm : (@ofLower α).symm = toLower := rfl
@[simp] lemma toLower_ofLower (a : WithLower α) : toLower (ofLower a) = a := rfl
@[simp] lemma ofLower_toLower (a : α) : ofLower (toLower a) = a := rfl | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | ofLower | `ofLower` is the identity function from the `WithLower` of a type. |
toLower_inj {a b : α} : toLower a = toLower b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | toLower_inj | null |
ofLower_inj {a b : WithLower α} : ofLower a = ofLower b ↔ a = b :=
Iff.rfl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | ofLower_inj | null |
@[elab_as_elim, cases_eliminator, induction_eliminator]
protected rec {β : WithLower α → Sort*} (h : ∀ a, β (toLower a)) : ∀ a, β a := fun a =>
h (ofLower a) | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | rec | A recursor for `WithLower`. Use as `induction x`. |
@[simp] toLower_le_toLower {x y : α} : toLower x ≤ toLower y ↔ x ≤ y := .rfl
@[simp] lemma toLower_lt_toLower {x y : α} : toLower x < toLower y ↔ x < y := .rfl
@[simp] lemma ofLower_le_ofLower {x y : WithLower α} : ofLower x ≤ ofLower y ↔ x ≤ y := .rfl
@[simp] lemma ofLower_lt_ofLower {x y : WithLower α} : ofLower x < ofLower y ↔ x < y := .rfl | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | toLower_le_toLower | null |
isOpen_preimage_ofLower : IsOpen (ofLower ⁻¹' s) ↔ IsOpen[lower α] s := Iff.rfl | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isOpen_preimage_ofLower | null |
isOpen_def (T : Set (WithLower α)) : IsOpen T ↔ IsOpen[lower α] (WithLower.toLower ⁻¹' T) :=
Iff.rfl | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isOpen_def | null |
continuous_toLower [TopologicalSpace α] [ClosedIciTopology α] :
Continuous (toLower : α → WithLower α) :=
continuous_generateFrom_iff.mpr <| by rintro _ ⟨a, rfl⟩; exact isClosed_Ici.isOpen_compl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | continuous_toLower | null |
WithUpper (α : Type*) := α | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | WithUpper | Type synonym for a preorder equipped with the upper topology. |
@[match_pattern] toUpper : α ≃ WithUpper α := Equiv.refl _ | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | toUpper | `toUpper` is the identity function to the `WithUpper` of a type. |
@[match_pattern] ofUpper : WithUpper α ≃ α := Equiv.refl _
@[simp] lemma toUpper_symm {α} : (@toUpper α).symm = ofUpper := rfl
@[simp] lemma ofUpper_symm : (@ofUpper α).symm = toUpper := rfl
@[simp] lemma toUpper_ofUpper (a : WithUpper α) : toUpper (ofUpper a) = a := rfl
@[simp] lemma ofUpper_toUpper (a : α) : ofUpper (toUpper a) = a := rfl | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | ofUpper | `ofUpper` is the identity function from the `WithUpper` of a type. |
toUpper_inj {a b : α} : toUpper a = toUpper b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | toUpper_inj | null |
ofUpper_inj {a b : WithUpper α} : ofUpper a = ofUpper b ↔ a = b := Iff.rfl | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | ofUpper_inj | null |
@[elab_as_elim, cases_eliminator, induction_eliminator]
protected rec {β : WithUpper α → Sort*} (h : ∀ a, β (toUpper a)) : ∀ a, β a := fun a =>
h (ofUpper a) | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | rec | A recursor for `WithUpper`. Use as `induction x`. |
@[simp] toUpper_le_toUpper {x y : α} : toUpper x ≤ toUpper y ↔ x ≤ y := .rfl
@[simp] lemma toUpper_lt_toUpper {x y : α} : toUpper x < toUpper y ↔ x < y := .rfl
@[simp] lemma ofUpper_le_ofUpper {x y : WithUpper α} : ofUpper x ≤ ofUpper y ↔ x ≤ y := .rfl
@[simp] lemma ofUpper_lt_ofUpper {x y : WithUpper α} : ofUpper x < ofUpper y ↔ x < y := .rfl | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | toUpper_le_toUpper | null |
isOpen_preimage_ofUpper : IsOpen (ofUpper ⁻¹' s) ↔ (upper α).IsOpen s := Iff.rfl | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isOpen_preimage_ofUpper | null |
isOpen_def {s : Set (WithUpper α)} : IsOpen s ↔ (upper α).IsOpen (toUpper ⁻¹' s) := Iff.rfl | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isOpen_def | null |
continuous_toUpper [TopologicalSpace α] [ClosedIicTopology α] :
Continuous (toUpper : α → WithUpper α) :=
continuous_generateFrom_iff.mpr <| by rintro _ ⟨a, rfl⟩; exact isClosed_Iic.isOpen_compl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | continuous_toUpper | null |
IsLower (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
topology_eq_lowerTopology : t = lower α
attribute [nolint docBlame] IsLower.topology_eq_lowerTopology | class | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | IsLower | The lower topology is the topology generated by the complements of the left-closed right-infinite
intervals. |
IsUpper (α : Type*) [t : TopologicalSpace α] [Preorder α] : Prop where
topology_eq_upperTopology : t = upper α
attribute [nolint docBlame] IsUpper.topology_eq_upperTopology | class | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | IsUpper | The upper topology is the topology generated by the complements of the right-closed left-infinite
intervals. |
WithLower.toDualHomeomorph [Preorder α] : WithLower α ≃ₜ WithUpper αᵒᵈ where
toFun := OrderDual.toDual
invFun := OrderDual.ofDual
left_inv := OrderDual.toDual_ofDual
right_inv := OrderDual.ofDual_toDual
continuous_toFun := continuous_coinduced_rng
continuous_invFun := continuous_coinduced_rng | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | WithLower.toDualHomeomorph | The lower topology is homeomorphic to the upper topology on the dual order |
lowerBasis (α : Type*) [Preorder α] :=
{ s : Set α | ∃ t : Set α, t.Finite ∧ (upperClosure t : Set α)ᶜ = s } | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | lowerBasis | The complements of the upper closures of finite sets are a collection of lower sets
which form a basis for the lower topology. |
topology_eq : ‹_› = lower α := topology_eq_lowerTopology
variable {α} | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | topology_eq | null |
withLowerHomeomorph : WithLower α ≃ₜ α :=
WithLower.ofLower.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | withLowerHomeomorph | If `α` is equipped with the lower topology, then it is homeomorphic to `WithLower α`. |
isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Ici a)ᶜ = t } s := by
rw [topology_eq α]; rfl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isOpen_iff_generate_Ici_compl | null |
_root_.OrderDual.instIsUpper [Preorder α] [TopologicalSpace α] [IsLower α] :
IsUpper αᵒᵈ where
topology_eq_upperTopology := topology_eq_lowerTopology (α := α) | instance | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | _root_.OrderDual.instIsUpper | null |
isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) := by
simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf]
exact h.isClosed_biUnion fun _ _ => isClosed_Ici | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isClosed_upperClosure | Left-closed right-infinite intervals [a, ∞) are closed in the lower topology. -/
instance : ClosedIciTopology α :=
⟨fun a ↦ isOpen_compl_iff.1 <| isOpen_iff_generate_Ici_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩⟩
/-- The upper closure of a finite set is closed in the lower topology. |
isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s := by
replace h := isOpen_iff_generate_Ici_compl.1 h
induction h with
| basic u h' => obtain ⟨a, rfl⟩ := h'; exact (isUpperSet_Ici a).compl
| univ => exact isLowerSet_univ
| inter u v _ _ hu2 hv2 => exact hu2.inter hv2
| sUnion _ _ ih => exact isLowerSet_sUnion ih | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isLowerSet_of_isOpen | Every set open in the lower topology is a lower set. |
isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s :=
isLowerSet_compl.1 <| isLowerSet_of_isOpen h.isOpen_compl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isUpperSet_of_isClosed | null |
tendsto_nhds_iff_not_le {β : Type*} {f : β → α} {l : Filter β} {x : α} :
Filter.Tendsto f l (𝓝 x) ↔ ∀ y, ¬y ≤ x → ∀ᶠ z in l, ¬y ≤ f z := by
simp [topology_eq_lowerTopology, tendsto_nhds_generateFrom_iff, Filter.Eventually, Ici,
compl_setOf] | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | tendsto_nhds_iff_not_le | null |
@[simp]
closure_singleton (a : α) : closure {a} = Ici a :=
Subset.antisymm ((closure_minimal fun _ h => h.ge) <| isClosed_Ici) <|
(isUpperSet_of_isClosed isClosed_closure).Ici_subset <| subset_closure rfl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | closure_singleton | The closure of a singleton `{a}` in the lower topology is the left-closed right-infinite interval
[a, ∞). |
protected isTopologicalBasis : IsTopologicalBasis (lowerBasis α) := by
convert isTopologicalBasis_of_subbasis (topology_eq α)
simp_rw [lowerBasis, coe_upperClosure, compl_iUnion]
ext s
constructor
· rintro ⟨F, hF, rfl⟩
refine ⟨(fun a => (Ici a)ᶜ) '' F, ⟨hF.image _, image_subset_iff.2 fun _ _ => ⟨_, rfl⟩⟩, ?_⟩
simp only [sInter_image]
· rintro ⟨F, ⟨hF, hs⟩, rfl⟩
haveI := hF.to_subtype
rw [subset_def, Subtype.forall'] at hs
choose f hf using hs
exact ⟨_, finite_range f, by simp_rw [biInter_range, hf, sInter_eq_iInter]⟩ | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isTopologicalBasis | null |
continuous_iff_Ici [TopologicalSpace β] {f : β → α} :
Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Ici a)) := by
obtain rfl := IsLower.topology_eq α
simp [continuous_generateFrom_iff] | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | continuous_iff_Ici | A function `f : β → α` with lower topology in the codomain is continuous
if and only if the preimage of every interval `Set.Ici a` is a closed set. |
upperBasis (α : Type*) [Preorder α] :=
{ s : Set α | ∃ t : Set α, t.Finite ∧ (lowerClosure t : Set α)ᶜ = s } | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | upperBasis | The lower topology on a partial order is T₀. -/
instance (priority := 90) t0Space : T0Space α :=
(t0Space_iff_inseparable α).2 fun x y h =>
Ici_injective <| by simpa only [inseparable_iff_closure_eq, closure_singleton] using h
end PartialOrder
section LinearOrder
variable [LinearOrder α] [TopologicalSpace α] [IsLower α]
lemma isTopologicalBasis_insert_univ_subbasis :
IsTopologicalBasis (insert univ {s : Set α | ∃ a, (Ici a)ᶜ = s}) :=
isTopologicalBasis_of_subbasis_of_inter (by rw [topology_eq α, lower]) (by
rintro _ ⟨b, rfl⟩ _ ⟨c, rfl⟩
use b ⊓ c
rw [compl_Ici, compl_Ici, compl_Ici, Iio_inter_Iio])
theorem tendsto_nhds_iff_lt {β : Type*} {f : β → α} {l : Filter β} {x : α} :
Filter.Tendsto f l (𝓝 x) ↔ ∀ y, x < y → ∀ᶠ z in l, f z < y := by
simp only [tendsto_nhds_iff_not_le, not_le]
end LinearOrder
section CompleteLinearOrder
variable [CompleteLinearOrder α] [t : TopologicalSpace α] [IsLower α]
lemma isTopologicalSpace_basis (U : Set α) : IsOpen U ↔ U = univ ∨ ∃ a, (Ici a)ᶜ = U := by
by_cases hU : U = univ
· simp only [hU, isOpen_univ, compl_Ici, true_or]
refine ⟨?_, isTopologicalBasis_insert_univ_subbasis.isOpen⟩
intro hO
apply Or.inr
convert IsTopologicalBasis.open_eq_sUnion isTopologicalBasis_insert_univ_subbasis hO
constructor
· intro ⟨a, ha⟩
use {U}
constructor
· apply subset_trans (singleton_subset_iff.mpr _) (subset_insert _ _)
use a
· rw [sUnion_singleton]
· intro ⟨S, hS1, hS2⟩
have hUS : univ ∉ S := by
by_contra hUS'
apply hU
rw [hS2]
exact sUnion_eq_univ_iff.mpr (fun a => ⟨univ, hUS', trivial⟩)
use sSup {a | (Ici a)ᶜ ∈ S}
rw [hS2, sUnion_eq_compl_sInter_compl, compl_inj_iff]
apply le_antisymm
· intro b hb
simp only [sInter_image, mem_iInter, mem_compl_iff]
intro s hs
obtain ⟨a, ha⟩ := (subset_insert_iff_of_notMem hUS).mp hS1 hs
subst hS2 ha
simp_all only [compl_Ici, mem_Ici, sSup_le_iff, mem_setOf_eq, mem_Iio, not_lt]
· intro b hb
rw [mem_Ici, sSup_le_iff]
intro c hc
simp only [sInter_image, mem_iInter] at hb
rw [← not_lt, ← mem_Iio, ← compl_Ici]
exact hb _ hc
end CompleteLinearOrder
end IsLower
namespace IsUpper
/-- The complements of the lower closures of finite sets are a collection of upper sets
which form a basis for the upper topology. |
topology_eq : ‹_› = upper α := topology_eq_upperTopology
variable {α} | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | topology_eq | null |
withUpperHomeomorph : WithUpper α ≃ₜ α :=
WithUpper.ofUpper.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ | def | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | withUpperHomeomorph | If `α` is equipped with the upper topology, then it is homeomorphic to `WithUpper α`. |
isOpen_iff_generate_Iic_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Iic a)ᶜ = t } s := by
rw [topology_eq α]; rfl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isOpen_iff_generate_Iic_compl | null |
_root_.OrderDual.instIsLower [Preorder α] [TopologicalSpace α] [IsUpper α] :
IsLower αᵒᵈ where
topology_eq_lowerTopology := topology_eq_upperTopology (α := α) | instance | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | _root_.OrderDual.instIsLower | null |
isClosed_lowerClosure (h : s.Finite) : IsClosed (lowerClosure s : Set α) :=
IsLower.isClosed_upperClosure (α := αᵒᵈ) h | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isClosed_lowerClosure | Left-infinite right-closed intervals (-∞,a] are closed in the upper topology. -/
instance : ClosedIicTopology α :=
⟨fun a ↦ isOpen_compl_iff.1 <| isOpen_iff_generate_Iic_compl.2 <| GenerateOpen.basic _ ⟨a, rfl⟩⟩
/-- The lower closure of a finite set is closed in the upper topology. |
isUpperSet_of_isOpen (h : IsOpen s) : IsUpperSet s :=
IsLower.isLowerSet_of_isOpen (α := αᵒᵈ) h | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isUpperSet_of_isOpen | Every set open in the upper topology is a upper set. |
isLowerSet_of_isClosed (h : IsClosed s) : IsLowerSet s :=
isUpperSet_compl.1 <| isUpperSet_of_isOpen h.isOpen_compl | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isLowerSet_of_isClosed | null |
tendsto_nhds_iff_not_le {β : Type*} {f : β → α} {l : Filter β} {x : α} :
Filter.Tendsto f l (𝓝 x) ↔ ∀ y, ¬x ≤ y → ∀ᶠ z in l, ¬f z ≤ y :=
IsLower.tendsto_nhds_iff_not_le (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | tendsto_nhds_iff_not_le | null |
@[simp]
closure_singleton (a : α) : closure {a} = Iic a :=
IsLower.closure_singleton (α := αᵒᵈ) _ | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | closure_singleton | The closure of a singleton `{a}` in the upper topology is the left-infinite right-closed interval
(-∞,a]. |
protected isTopologicalBasis : IsTopologicalBasis (upperBasis α) :=
IsLower.isTopologicalBasis (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isTopologicalBasis | null |
continuous_iff_Iic [TopologicalSpace β] {f : β → α} :
Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Iic a)) :=
IsLower.continuous_iff_Ici (α := αᵒᵈ) | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | continuous_iff_Iic | A function `f : β → α` with upper topology in the codomain is continuous
if and only if the preimage of every interval `Set.Iic a` is a closed set. |
isTopologicalBasis_insert_univ_subbasis :
IsTopologicalBasis (insert univ {s : Set α | ∃ a, (Iic a)ᶜ = s}) :=
IsLower.isTopologicalBasis_insert_univ_subbasis (α := αᵒᵈ) | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isTopologicalBasis_insert_univ_subbasis | null |
tendsto_nhds_iff_lt {β : Type*} {f : β → α} {l : Filter β} {x : α} :
Filter.Tendsto f l (𝓝 x) ↔ ∀ y < x, ∀ᶠ z in l, y < f z :=
IsLower.tendsto_nhds_iff_lt (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | tendsto_nhds_iff_lt | null |
isTopologicalSpace_basis (U : Set α) : IsOpen U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U :=
IsLower.isTopologicalSpace_basis (α := αᵒᵈ) U | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isTopologicalSpace_basis | null |
instIsLowerProd [Preorder α] [TopologicalSpace α] [IsLower α]
[OrderBot α] [Preorder β] [TopologicalSpace β] [IsLower β] [OrderBot β] :
IsLower (α × β) where
topology_eq_lowerTopology := by
refine le_antisymm (le_generateFrom ?_) ?_
· rintro _ ⟨x, rfl⟩
exact (isClosed_Ici.prod isClosed_Ici).isOpen_compl
rw [(IsLower.isTopologicalBasis.prod
IsLower.isTopologicalBasis).eq_generateFrom, le_generateFrom_iff_subset_isOpen,
image2_subset_iff]
rintro _ ⟨s, hs, rfl⟩ _ ⟨t, ht, rfl⟩
dsimp
simp_rw [coe_upperClosure, compl_iUnion, prod_eq, preimage_iInter, preimage_compl]
let _ : TopologicalSpace (α × β) := lower (α × β)
refine (hs.isOpen_biInter fun a _ => ?_).inter (ht.isOpen_biInter fun b _ => ?_)
· exact GenerateOpen.basic _ ⟨(a, ⊥), by simp [Ici_prod_eq, prod_univ]⟩
· exact GenerateOpen.basic _ ⟨(⊥, b), by simp [Ici_prod_eq, univ_prod]⟩ | instance | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | instIsLowerProd | null |
instIsUpperProd [Preorder α] [TopologicalSpace α] [IsUpper α]
[OrderTop α] [Preorder β] [TopologicalSpace β] [IsUpper β] [OrderTop β] :
IsUpper (α × β) where
topology_eq_upperTopology := by
suffices IsLower (α × β)ᵒᵈ from IsLower.topology_eq_lowerTopology (α := (α × β)ᵒᵈ)
exact instIsLowerProd (α := αᵒᵈ) (β := βᵒᵈ) | instance | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | instIsUpperProd | null |
protected _root_.sInfHom.continuous (f : sInfHom α β) : Continuous f := by
refine IsLower.continuous_iff_Ici.2 fun b => ?_
convert isClosed_Ici (a := sInf <| f ⁻¹' Ici b)
refine Subset.antisymm (fun a => sInf_le) fun a ha => le_trans ?_ <|
OrderHomClass.mono (f : α →o β) ha
refine LE.le.trans ?_ (map_sInf f _).ge
simp | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | _root_.sInfHom.continuous | null |
protected _root_.sSupHom.continuous (f : sSupHom α β) : Continuous f :=
sInfHom.continuous (α := αᵒᵈ) (β := βᵒᵈ) (sSupHom.dual.toFun f) | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | _root_.sSupHom.continuous | null |
isUpper_orderDual [Preorder α] [TopologicalSpace α] : IsUpper αᵒᵈ ↔ IsLower α := by
constructor
· apply OrderDual.instIsLower
· apply OrderDual.instIsUpper | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isUpper_orderDual | null |
isLower_orderDual [Preorder α] [TopologicalSpace α] : IsLower αᵒᵈ ↔ IsUpper α :=
isUpper_orderDual.symm | lemma | Topology | [
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Order.Lattice"
] | Mathlib/Topology/Order/LowerUpperTopology.lean | isLower_orderDual | null |
MonotoneOn.insert_of_continuousWithinAt [TopologicalSpace β] [OrderClosedTopology β]
(hf : MonotoneOn f s) (hx : ClusterPt x (𝓟 s)) (h'x : ContinuousWithinAt f s x) :
MonotoneOn f (insert x s) := by
have : (𝓝[s] x).NeBot := hx
apply monotoneOn_insert_iff.2 ⟨fun b hb hbx ↦ ?_, fun b hb hxb ↦ ?_, hf⟩
· rcases hbx.eq_or_lt with rfl | hbx
· exact le_rfl
simp only [ContinuousWithinAt] at h'x
apply ge_of_tendsto h'x
have : s ∩ Ioi b ∈ 𝓝[s] x := inter_mem_nhdsWithin _ (Ioi_mem_nhds hbx)
filter_upwards [this] with y hy using hf hb hy.1 (le_of_lt hy.2)
· rcases hxb.eq_or_lt with rfl | hxb
· exact le_rfl
simp only [ContinuousWithinAt] at h'x
apply le_of_tendsto h'x
have : s ∩ Iio b ∈ 𝓝[s] x := inter_mem_nhdsWithin _ (Iio_mem_nhds hxb)
filter_upwards [this] with y hy
exact hf hy.1 hb (le_of_lt hy.2) | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.insert_of_continuousWithinAt | null |
MonotoneOn.countable_setOf_two_preimages [SecondCountableTopology α]
(hf : MonotoneOn f s) :
Set.Countable {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c} := by
nontriviality α
let t := {c | ∃ x, ∃ y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}
have : ∀ c ∈ t, ∃ x, ∃ y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c := fun c hc ↦ hc
choose! x y hxs hys hxy hfx hfy using this
let u := x '' t
suffices H : Set.Countable (x '' t) by
have : Set.InjOn x t := by
intro c hc d hd hcd
have : f (x c) = f (x d) := by simp [hcd]
rwa [hfx _ hc, hfx _ hd] at this
exact countable_of_injective_of_countable_image this H
apply Set.PairwiseDisjoint.countable_of_Ioo (y := fun a ↦ y (f a)); swap
· rintro a ⟨c, hc, rfl⟩
rw [hfx _ hc]
exact hxy _ hc
simp only [PairwiseDisjoint, Set.Pairwise, mem_image, onFun, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
intro c hc d hd hcd
wlog H : c < d generalizing c d with h
· apply (h d hd c hc hcd.symm ?_).symm
have : c ≠ d := fun h ↦ hcd (congrArg x h)
order
simp only [disjoint_iff_forall_ne, mem_Ioo, ne_eq, and_imp]
rintro a xca ayc b xda ayd rfl
rw [hfx _ hc] at ayc
have : x d ≤ y c := (xda.trans ayc).le
have : f (x d) ≤ f (y c) := hf (hxs _ hd) (hys _ hc) this
rw [hfx _ hd, hfy _ hc] at this
exact not_le.2 H this | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.countable_setOf_two_preimages | If a function is monotone on a set in a second countable topological space, then there
are only countably many points that have several preimages. |
Monotone.countable_setOf_two_preimages [SecondCountableTopology α]
(hf : Monotone f) :
Set.Countable {c | ∃ x y, x < y ∧ f x = c ∧ f y = c} := by
rw [← monotoneOn_univ] at hf
simpa using hf.countable_setOf_two_preimages | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.countable_setOf_two_preimages | If a function is monotone in a second countable topological space, then there
are only countably many points that have several preimages. |
AntitoneOn.countable_setOf_two_preimages [SecondCountableTopology α]
(hf : AntitoneOn f s) :
Set.Countable {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c} :=
(MonotoneOn.countable_setOf_two_preimages hf.dual_right :) | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | AntitoneOn.countable_setOf_two_preimages | If a function is antitone on a set in a second countable topological space, then there
are only countably many points that have several preimages. |
Antitone.countable_setOf_two_preimages [SecondCountableTopology α]
(hf : Antitone f) :
Set.Countable {c | ∃ x y, x < y ∧ f x = c ∧ f y = c} :=
(Monotone.countable_setOf_two_preimages hf.dual_right :) | lemma | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.countable_setOf_two_preimages | If a function is antitone in a second countable topological space, then there
are only countably many points that have several preimages. |
MonotoneOn.countable_not_continuousWithinAt_Ioi (hf : MonotoneOn f s) :
Set.Countable {x ∈ s | ¬ContinuousWithinAt f (s ∩ Ioi x) x} := by
apply (countable_image_lt_image_Ioi_within s f).mono
rintro x ⟨xs, hx : ¬ContinuousWithinAt f (s ∩ Ioi x) x⟩
dsimp only [mem_setOf_eq]
contrapose! hx
refine tendsto_order.2 ⟨fun m hm => ?_, fun u hu => ?_⟩
· filter_upwards [@self_mem_nhdsWithin _ _ x (s ∩ Ioi x)] with y hy
exact hm.trans_le (hf xs hy.1 (le_of_lt hy.2))
rcases hx xs u hu with ⟨v, vs, xv, fvu⟩
have : s ∩ Ioo x v ∈ 𝓝[s ∩ Ioi x] x := by simp [nhdsWithin_inter, mem_inf_of_left,
self_mem_nhdsWithin, mem_inf_of_right, Ioo_mem_nhdsGT xv]
filter_upwards [this] with y hy
exact (hf hy.1 vs hy.2.2.le).trans_lt fvu | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.countable_not_continuousWithinAt_Ioi | In a second countable space, the set of points where a monotone function is not right-continuous
within a set is at most countable. Superseded by `MonotoneOn.countable_not_continuousWithinAt`
which gives the two-sided version. |
MonotoneOn.countable_not_continuousWithinAt_Iio (hf : MonotoneOn f s) :
Set.Countable {x ∈ s | ¬ContinuousWithinAt f (s ∩ Iio x) x} :=
hf.dual.countable_not_continuousWithinAt_Ioi | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.countable_not_continuousWithinAt_Iio | In a second countable space, the set of points where a monotone function is not left-continuous
within a set is at most countable. Superseded by `MonotoneOn.countable_not_continuousWithinAt`
which gives the two-sided version. |
MonotoneOn.countable_not_continuousWithinAt (hf : MonotoneOn f s) :
Set.Countable {x ∈ s | ¬ContinuousWithinAt f s x} := by
apply (hf.countable_not_continuousWithinAt_Ioi.union hf.countable_not_continuousWithinAt_Iio).mono
refine compl_subset_compl.1 ?_
simp only [compl_union]
rintro x ⟨hx, h'x⟩
simp only [mem_compl_iff, mem_setOf_eq, not_and, not_not] at hx h'x ⊢
intro xs
exact continuousWithinAt_iff_continuous_left'_right'.2 ⟨h'x xs, hx xs⟩ | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.countable_not_continuousWithinAt | In a second countable space, the set of points where a monotone function is not continuous
within a set is at most countable. |
Monotone.countable_not_continuousAt (hf : Monotone f) :
Set.Countable {x | ¬ContinuousAt f x} := by
simpa [continuousWithinAt_univ] using (hf.monotoneOn univ).countable_not_continuousWithinAt | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.countable_not_continuousAt | In a second countable space, the set of points where a monotone function is not continuous
is at most countable. |
_root_.AntitoneOn.countable_not_continuousWithinAt
{s : Set α} (hf : AntitoneOn f s) :
Set.Countable {x ∈ s | ¬ContinuousWithinAt f s x} :=
hf.dual_right.countable_not_continuousWithinAt | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | _root_.AntitoneOn.countable_not_continuousWithinAt | In a second countable space, the set of points where an antitone function is not continuous
within a set is at most countable. |
Antitone.countable_not_continuousAt (hf : Antitone f) :
Set.Countable {x | ¬ContinuousAt f x} :=
hf.dual_right.countable_not_continuousAt | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.countable_not_continuousAt | In a second countable space, the set of points where an antitone function is not continuous
is at most countable. |
MonotoneOn.map_csSup_of_continuousWithinAt {f : α → β} {A : Set α}
(Cf : ContinuousWithinAt f A (sSup A))
(Mf : MonotoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
f (sSup A) = sSup (f '' A) :=
.symm <| ((isLUB_csSup A_nonemp A_bdd).isLUB_of_tendsto Mf A_nonemp <|
Cf.mono_left fun ⦃_⦄ a ↦ a).csSup_eq (A_nonemp.image f) | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.map_csSup_of_continuousWithinAt | A monotone function continuous at the supremum of a nonempty set sends this supremum to
the supremum of the image of this set. |
Monotone.map_csSup_of_continuousAt {f : α → β} {A : Set α}
(Cf : ContinuousAt f (sSup A)) (Mf : Monotone f) (A_nonemp : A.Nonempty)
(A_bdd : BddAbove A := by bddDefault) : f (sSup A) = sSup (f '' A) :=
MonotoneOn.map_csSup_of_continuousWithinAt Cf.continuousWithinAt
(Mf.monotoneOn _) A_nonemp A_bdd | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.map_csSup_of_continuousAt | A monotone function continuous at the supremum of a nonempty set sends this supremum to
the supremum of the image of this set. |
Monotone.map_ciSup_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f)
(bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨆ i, f (g i) := by
rw [iSup, Monotone.map_csSup_of_continuousAt Cf Mf (range_nonempty g) bdd, ← range_comp, iSup,
comp_def] | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.map_ciSup_of_continuousAt | A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
supremum to the indexed supremum of the composition. |
MonotoneOn.map_csInf_of_continuousWithinAt {f : α → β} {A : Set α}
(Cf : ContinuousWithinAt f A (sInf A))
(Mf : MonotoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
f (sInf A) = sInf (f '' A) :=
MonotoneOn.map_csSup_of_continuousWithinAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual A_nonemp A_bdd | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.map_csInf_of_continuousWithinAt | A monotone function continuous at the infimum of a nonempty set sends this infimum to
the infimum of the image of this set. |
Monotone.map_csInf_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
(Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
f (sInf A) = sInf (f '' A) :=
Monotone.map_csSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual A_nonemp A_bdd | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.map_csInf_of_continuousAt | A monotone function continuous at the infimum of a nonempty set sends this infimum to
the infimum of the image of this set. |
Monotone.map_ciInf_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
(bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨅ i, f (g i) := by
rw [iInf, Monotone.map_csInf_of_continuousAt Cf Mf (range_nonempty g) bdd, ← range_comp, iInf,
comp_def] | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.map_ciInf_of_continuousAt | A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed infimum of the composition. |
AntitoneOn.map_csInf_of_continuousWithinAt {f : α → β} {A : Set α}
(Cf : ContinuousWithinAt f A (sInf A))
(Af : AntitoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
f (sInf A) = sSup (f '' A) :=
MonotoneOn.map_csInf_of_continuousWithinAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | AntitoneOn.map_csInf_of_continuousWithinAt | An antitone function continuous at the infimum of a nonempty set sends this infimum to
the supremum of the image of this set. |
Antitone.map_csInf_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
(Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
f (sInf A) = sSup (f '' A) :=
Monotone.map_csInf_of_continuousAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.map_csInf_of_continuousAt | An antitone function continuous at the infimum of a nonempty set sends this infimum to
the supremum of the image of this set. |
Antitone.map_ciInf_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Af : Antitone f)
(bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨆ i, f (g i) := by
rw [iInf, Antitone.map_csInf_of_continuousAt Cf Af (range_nonempty g) bdd, ← range_comp, iSup,
comp_def] | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.map_ciInf_of_continuousAt | An antitone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed supremum of the composition. |
AntitoneOn.map_csSup_of_continuousWithinAt {f : α → β} {A : Set α}
(Cf : ContinuousWithinAt f A (sSup A))
(Af : AntitoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
f (sSup A) = sInf (f '' A) :=
MonotoneOn.map_csSup_of_continuousWithinAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | AntitoneOn.map_csSup_of_continuousWithinAt | An antitone function continuous at the supremum of a nonempty set sends this supremum to
the infimum of the image of this set. |
Antitone.map_csSup_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A))
(Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
f (sSup A) = sInf (f '' A) :=
Monotone.map_csSup_of_continuousAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.map_csSup_of_continuousAt | An antitone function continuous at the supremum of a nonempty set sends this supremum to
the infimum of the image of this set. |
Antitone.map_ciSup_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Af : Antitone f)
(bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨅ i, f (g i) := by
rw [iSup, Antitone.map_csSup_of_continuousAt Cf Af (range_nonempty g) bdd, ← range_comp, iInf,
comp_def] | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.map_ciSup_of_continuousAt | An antitone function continuous at the indexed supremum over a nonempty `Sort` sends this
indexed supremum to the indexed infimum of the composition. |
sSup_mem_closure {s : Set α} (hs : s.Nonempty) : sSup s ∈ closure s :=
(isLUB_sSup s).mem_closure hs | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | sSup_mem_closure | null |
sInf_mem_closure {s : Set α} (hs : s.Nonempty) : sInf s ∈ closure s :=
(isGLB_sInf s).mem_closure hs | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | sInf_mem_closure | null |
IsClosed.sSup_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sSup s ∈ s :=
(isLUB_sSup s).mem_of_isClosed hs hc | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | IsClosed.sSup_mem | null |
IsClosed.sInf_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sInf s ∈ s :=
(isGLB_sInf s).mem_of_isClosed hs hc | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | IsClosed.sInf_mem | null |
MonotoneOn.map_sSup_of_continuousWithinAt {f : α → β} {s : Set α}
(Cf : ContinuousWithinAt f s (sSup s))
(Mf : MonotoneOn f s) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) := by
rcases s.eq_empty_or_nonempty with h | h
· simp [h, fbot]
· exact Mf.map_csSup_of_continuousWithinAt Cf h | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.map_sSup_of_continuousWithinAt | A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. |
Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) :=
MonotoneOn.map_sSup_of_continuousWithinAt Cf.continuousWithinAt (Mf.monotoneOn _) fbot | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.map_sSup_of_continuousAt | A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. |
Monotone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fbot : f ⊥ = ⊥) :
f (⨆ i, g i) = ⨆ i, f (g i) := by
rw [iSup, Mf.map_sSup_of_continuousAt Cf fbot, ← range_comp, iSup, comp_def] | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.map_iSup_of_continuousAt | If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. |
MonotoneOn.map_sInf_of_continuousWithinAt {f : α → β} {s : Set α}
(Cf : ContinuousWithinAt f s (sInf s)) (Mf : MonotoneOn f s) (ftop : f ⊤ = ⊤) :
f (sInf s) = sInf (f '' s) :=
MonotoneOn.map_sSup_of_continuousWithinAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | MonotoneOn.map_sInf_of_continuousWithinAt | A monotone function `f` sending `top` to `top` and continuous at the infimum of a set sends
this infimum to the infimum of the image of this set. |
Monotone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
(Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (sInf s) = sInf (f '' s) :=
Monotone.map_sSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.map_sInf_of_continuousAt | A monotone function `f` sending `top` to `top` and continuous at the infimum of a set sends
this infimum to the infimum of the image of this set. |
Monotone.map_iInf_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (iInf g) = iInf (f ∘ g) :=
Monotone.map_iSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Monotone.map_iInf_of_continuousAt | If a monotone function sending `top` to `top` is continuous at the indexed infimum over
a `Sort`, then it sends this indexed infimum to the indexed infimum of the composition. |
AntitoneOn.map_sSup_of_continuousWithinAt {f : α → β} {s : Set α}
(Cf : ContinuousWithinAt f s (sSup s)) (Af : AntitoneOn f s) (fbot : f ⊥ = ⊤) :
f (sSup s) = sInf (f '' s) :=
MonotoneOn.map_sSup_of_continuousWithinAt
(show ContinuousWithinAt (OrderDual.toDual ∘ f) s (sSup s) from Cf) Af fbot | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | AntitoneOn.map_sSup_of_continuousWithinAt | An antitone function `f` sending `bot` to `top` and continuous at the supremum of a set sends
this supremum to the infimum of the image of this set. |
Antitone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Af : Antitone f) (fbot : f ⊥ = ⊤) : f (sSup s) = sInf (f '' s) :=
Monotone.map_sSup_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (sSup s) from Cf) Af
fbot | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.map_sSup_of_continuousAt | An antitone function `f` sending `bot` to `top` and continuous at the supremum of a set sends
this supremum to the infimum of the image of this set. |
Antitone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Af : Antitone f) (fbot : f ⊥ = ⊤) :
f (⨆ i, g i) = ⨅ i, f (g i) :=
Monotone.map_iSup_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (iSup g) from Cf) Af
fbot | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.map_iSup_of_continuousAt | An antitone function sending `bot` to `top` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. |
AntitoneOn.map_sInf_of_continuousWithinAt {f : α → β} {s : Set α}
(Cf : ContinuousWithinAt f s (sInf s)) (Af : AntitoneOn f s) (ftop : f ⊤ = ⊥) :
f (sInf s) = sSup (f '' s) :=
MonotoneOn.map_sInf_of_continuousWithinAt
(show ContinuousWithinAt (OrderDual.toDual ∘ f) s (sInf s) from Cf) Af ftop | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | AntitoneOn.map_sInf_of_continuousWithinAt | An antitone function `f` sending `top` to `bot` and continuous at the infimum of a set sends
this infimum to the supremum of the image of this set. |
Antitone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
(Af : Antitone f) (ftop : f ⊤ = ⊥) : f (sInf s) = sSup (f '' s) :=
Monotone.map_sInf_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (sInf s) from Cf) Af
ftop | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.map_sInf_of_continuousAt | An antitone function `f` sending `top` to `bot` and continuous at the infimum of a set sends
this infimum to the supremum of the image of this set. |
Antitone.map_iInf_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Af : Antitone f) (ftop : f ⊤ = ⊥) : f (iInf g) = iSup (f ∘ g) :=
Monotone.map_iInf_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (iInf g) from Cf) Af
ftop | theorem | Topology | [
"Mathlib.Tactic.Order",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/Monotone.lean | Antitone.map_iInf_of_continuousAt | If an antitone function sending `top` to `bot` is continuous at the indexed infimum over
a `Sort`, then it sends this indexed infimum to the indexed supremum of the composition. |
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