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 ⌀ |
|---|---|---|---|---|---|---|
isPreconnected_Ico : IsPreconnected (Ico a b) :=
ordConnected_Ico.isPreconnected | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isPreconnected_Ico | null |
isConnected_Ici : IsConnected (Ici a) :=
⟨nonempty_Ici, isPreconnected_Ici⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isConnected_Ici | null |
isConnected_Iic : IsConnected (Iic a) :=
⟨nonempty_Iic, isPreconnected_Iic⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isConnected_Iic | null |
isConnected_Ioi [NoMaxOrder α] : IsConnected (Ioi a) :=
⟨nonempty_Ioi, isPreconnected_Ioi⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isConnected_Ioi | null |
isConnected_Iio [NoMinOrder α] : IsConnected (Iio a) :=
⟨nonempty_Iio, isPreconnected_Iio⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isConnected_Iio | null |
isConnected_Icc (h : a ≤ b) : IsConnected (Icc a b) :=
⟨nonempty_Icc.2 h, isPreconnected_Icc⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isConnected_Icc | null |
isConnected_Ioo (h : a < b) : IsConnected (Ioo a b) :=
⟨nonempty_Ioo.2 h, isPreconnected_Ioo⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isConnected_Ioo | null |
isConnected_Ioc (h : a < b) : IsConnected (Ioc a b) :=
⟨nonempty_Ioc.2 h, isPreconnected_Ioc⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isConnected_Ioc | null |
isConnected_Ico (h : a < b) : IsConnected (Ico a b) :=
⟨nonempty_Ico.2 h, isPreconnected_Ico⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isConnected_Ico | null |
setOf_isPreconnected_eq_of_ordered :
{ s : Set α | IsPreconnected s } =
range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
(range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) := by
refine Subset.antisymm setOf_isPreconnected_subset_of_ordered ?_
simp only [subset_def, forall_mem_range, uncurry, or_imp, forall_and, mem_union,
mem_setOf_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true,
isPreconnected_Icc, isPreconnected_Ico, isPreconnected_Ioc, isPreconnected_Ioo,
isPreconnected_Ioi, isPreconnected_Iio, isPreconnected_Ici, isPreconnected_Iic,
isPreconnected_univ, isPreconnected_empty] | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | setOf_isPreconnected_eq_of_ordered | In a dense conditionally complete linear order, the set of preconnected sets is exactly
the set of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, `(-∞, +∞)`,
or `∅`. Though one can represent `∅` as `(sInf s, sInf s)`, we include it into the list of
possible cases to improve readability. |
isTotallyDisconnected_iff_lt {s : Set α} :
IsTotallyDisconnected s ↔ ∀ x ∈ s, ∀ y ∈ s, x < y → ∃ z ∉ s, z ∈ Ioo x y := by
simp only [IsTotallyDisconnected, isPreconnected_iff_ordConnected, ← not_nontrivial_iff,
nontrivial_iff_exists_lt, not_exists, not_and]
refine ⟨fun h x hx y hy hxy ↦ ?_, fun h t hts ht x hx y hy hxy ↦ ?_⟩
· simp_rw [← not_ordConnected_inter_Icc_iff hx hy]
exact fun hs ↦ h _ inter_subset_left hs _ ⟨hx, le_rfl, hxy.le⟩ _ ⟨hy, hxy.le, le_rfl⟩ hxy
· obtain ⟨z, h1z, h2z⟩ := h x (hts hx) y (hts hy) hxy
exact h1z <| hts <| ht.1 hx hy ⟨h2z.1.le, h2z.2.le⟩
/-! | lemma | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | isTotallyDisconnected_iff_lt | This lemmas characterizes when a subset `s` of a densely ordered conditionally complete linear
order is totally disconnected with respect to the order topology: between any two distinct points
of `s` must lie a point not in `s`. |
intermediate_value_Icc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
Icc (f a) (f b) ⊆ f '' Icc a b :=
isPreconnected_Icc.intermediate_value (left_mem_Icc.2 hab) (right_mem_Icc.2 hab) hf | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_Icc | **Intermediate Value Theorem** for continuous functions on closed intervals, case
`f a ≤ t ≤ f b`. |
intermediate_value_Icc' {a b : α} (hab : a ≤ b) {f : α → δ}
(hf : ContinuousOn f (Icc a b)) : Icc (f b) (f a) ⊆ f '' Icc a b :=
isPreconnected_Icc.intermediate_value (right_mem_Icc.2 hab) (left_mem_Icc.2 hab) hf | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_Icc' | **Intermediate Value Theorem** for continuous functions on closed intervals, case
`f a ≥ t ≥ f b`. |
intermediate_value_uIcc {a b : α} {f : α → δ} (hf : ContinuousOn f [[a, b]]) :
[[f a, f b]] ⊆ f '' uIcc a b := by
cases le_total (f a) (f b) <;> simp [*, isPreconnected_uIcc.intermediate_value] | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_uIcc | **Intermediate Value Theorem** for continuous functions on closed intervals, unordered case. |
exists_mem_uIcc_isFixedPt {a b : α} {f : α → α} (hf : ContinuousOn f (uIcc a b))
(ha : a ≤ f a) (hb : f b ≤ b) : ∃ c ∈ [[a, b]], IsFixedPt f c :=
isPreconnected_uIcc.intermediate_value₂ right_mem_uIcc left_mem_uIcc hf continuousOn_id hb ha | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | exists_mem_uIcc_isFixedPt | If `f : α → α` is continuous on `[[a, b]]`, `a ≤ f a`, and `f b ≤ b`,
then `f` has a fixed point on `[[a, b]]`. |
exists_mem_Icc_isFixedPt {a b : α} {f : α → α} (hf : ContinuousOn f (Icc a b))
(hle : a ≤ b) (ha : a ≤ f a) (hb : f b ≤ b) : ∃ c ∈ Icc a b, IsFixedPt f c :=
isPreconnected_Icc.intermediate_value₂
(right_mem_Icc.2 hle) (left_mem_Icc.2 hle) hf continuousOn_id hb ha | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | exists_mem_Icc_isFixedPt | If `f : α → α` is continuous on `[a, b]`, `a ≤ b`, `a ≤ f a`, and `f b ≤ b`,
then `f` has a fixed point on `[a, b]`.
In particular, if `[a, b]` is forward-invariant under `f`,
then `f` has a fixed point on `[a, b]`, see `exists_mem_Icc_isFixedPt_of_mapsTo`. |
exists_mem_Icc_isFixedPt_of_mapsTo {a b : α} {f : α → α} (hf : ContinuousOn f (Icc a b))
(hle : a ≤ b) (hmaps : MapsTo f (Icc a b) (Icc a b)) : ∃ c ∈ Icc a b, IsFixedPt f c :=
exists_mem_Icc_isFixedPt hf hle (hmaps <| left_mem_Icc.2 hle).1 (hmaps <| right_mem_Icc.2 hle).2 | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | exists_mem_Icc_isFixedPt_of_mapsTo | If a closed interval is forward-invariant under a continuous map `f : α → α`,
then this map has a fixed point on this interval. |
intermediate_value_Ico {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
Ico (f a) (f b) ⊆ f '' Ico a b :=
Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_lt_of_ge (he ▸ h.1))) fun hlt =>
@IsPreconnected.intermediate_value_Ico _ _ _ _ _ _ _ isPreconnected_Ico _ _ ⟨refl a, hlt⟩
(right_nhdsWithin_Ico_neBot hlt) inf_le_right _ (hf.mono Ico_subset_Icc_self) _
((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ico_subset_Icc_self) | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_Ico | null |
intermediate_value_Ico' {a b : α} (hab : a ≤ b) {f : α → δ}
(hf : ContinuousOn f (Icc a b)) : Ioc (f b) (f a) ⊆ f '' Ico a b :=
Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_lt_of_ge (he ▸ h.2))) fun hlt =>
@IsPreconnected.intermediate_value_Ioc _ _ _ _ _ _ _ isPreconnected_Ico _ _ ⟨refl a, hlt⟩
(right_nhdsWithin_Ico_neBot hlt) inf_le_right _ (hf.mono Ico_subset_Icc_self) _
((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ico_subset_Icc_self) | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_Ico' | null |
intermediate_value_Ioc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
Ioc (f a) (f b) ⊆ f '' Ioc a b :=
Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_le_of_gt (he ▸ h.1))) fun hlt =>
@IsPreconnected.intermediate_value_Ioc _ _ _ _ _ _ _ isPreconnected_Ioc _ _ ⟨hlt, refl b⟩
(left_nhdsWithin_Ioc_neBot hlt) inf_le_right _ (hf.mono Ioc_subset_Icc_self) _
((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioc_subset_Icc_self) | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_Ioc | null |
intermediate_value_Ioc' {a b : α} (hab : a ≤ b) {f : α → δ}
(hf : ContinuousOn f (Icc a b)) : Ico (f b) (f a) ⊆ f '' Ioc a b :=
Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_le_of_gt (he ▸ h.2))) fun hlt =>
@IsPreconnected.intermediate_value_Ico _ _ _ _ _ _ _ isPreconnected_Ioc _ _ ⟨hlt, refl b⟩
(left_nhdsWithin_Ioc_neBot hlt) inf_le_right _ (hf.mono Ioc_subset_Icc_self) _
((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioc_subset_Icc_self) | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_Ioc' | null |
intermediate_value_Ioo {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
Ioo (f a) (f b) ⊆ f '' Ioo a b :=
Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_lt_of_gt (he ▸ h.1))) fun hlt =>
@IsPreconnected.intermediate_value_Ioo _ _ _ _ _ _ _ isPreconnected_Ioo _ _
(left_nhdsWithin_Ioo_neBot hlt) (right_nhdsWithin_Ioo_neBot hlt) inf_le_right inf_le_right _
(hf.mono Ioo_subset_Icc_self) _ _
((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioo_subset_Icc_self)
((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ioo_subset_Icc_self) | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_Ioo | null |
intermediate_value_Ioo' {a b : α} (hab : a ≤ b) {f : α → δ}
(hf : ContinuousOn f (Icc a b)) : Ioo (f b) (f a) ⊆ f '' Ioo a b :=
Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_lt_of_gt (he ▸ h.2))) fun hlt =>
@IsPreconnected.intermediate_value_Ioo _ _ _ _ _ _ _ isPreconnected_Ioo _ _
(right_nhdsWithin_Ioo_neBot hlt) (left_nhdsWithin_Ioo_neBot hlt) inf_le_right inf_le_right _
(hf.mono Ioo_subset_Icc_self) _ _
((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ioo_subset_Icc_self)
((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioo_subset_Icc_self) | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | intermediate_value_Ioo' | null |
ContinuousOn.surjOn_Icc {s : Set α} [hs : OrdConnected s] {f : α → δ}
(hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : SurjOn f s (Icc (f a) (f b)) :=
hs.isPreconnected.intermediate_value ha hb hf | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | ContinuousOn.surjOn_Icc | **Intermediate value theorem**: if `f` is continuous on an order-connected set `s` and `a`,
`b` are two points of this set, then `f` sends `s` to a superset of `Icc (f x) (f y)`. |
ContinuousOn.surjOn_uIcc {s : Set α} [hs : OrdConnected s] {f : α → δ}
(hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
SurjOn f s (uIcc (f a) (f b)) := by
rcases le_total (f a) (f b) with hab | hab <;> simp [hf.surjOn_Icc, *] | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | ContinuousOn.surjOn_uIcc | **Intermediate value theorem**: if `f` is continuous on an order-connected set `s` and `a`,
`b` are two points of this set, then `f` sends `s` to a superset of `[f x, f y]`. |
Continuous.surjective {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atTop atTop)
(h_bot : Tendsto f atBot atBot) : Function.Surjective f := fun p =>
mem_range_of_exists_le_of_exists_ge hf (h_bot.eventually (eventually_le_atBot p)).exists
(h_top.eventually (eventually_ge_atTop p)).exists | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | Continuous.surjective | A continuous function which tendsto `Filter.atTop` along `Filter.atTop` and to `atBot` along
`at_bot` is surjective. |
Continuous.surjective' {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atBot atTop)
(h_bot : Tendsto f atTop atBot) : Function.Surjective f :=
Continuous.surjective (α := αᵒᵈ) hf h_top h_bot | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | Continuous.surjective' | A continuous function which tendsto `Filter.atBot` along `Filter.atTop` and to `Filter.atTop`
along `atBot` is surjective. |
ContinuousOn.surjOn_of_tendsto {f : α → δ} {s : Set α} [OrdConnected s] (hs : s.Nonempty)
(hf : ContinuousOn f s) (hbot : Tendsto (fun x : s => f x) atBot atBot)
(htop : Tendsto (fun x : s => f x) atTop atTop) : SurjOn f s univ :=
haveI := Classical.inhabited_of_nonempty hs.to_subtype
surjOn_iff_surjective.2 <| hf.restrict.surjective htop hbot | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | ContinuousOn.surjOn_of_tendsto | If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s`
tends to `at_bot : Filter β` along `at_bot : Filter ↥s` and tends to `Filter.atTop : Filter β` along
`Filter.atTop : Filter ↥s`, then the restriction of `f` to `s` is surjective. We formulate the
conclusion as `Function.surjOn f s Set.univ`. |
ContinuousOn.surjOn_of_tendsto' {f : α → δ} {s : Set α} [OrdConnected s] (hs : s.Nonempty)
(hf : ContinuousOn f s) (hbot : Tendsto (fun x : s => f x) atBot atTop)
(htop : Tendsto (fun x : s => f x) atTop atBot) : SurjOn f s univ :=
ContinuousOn.surjOn_of_tendsto (δ := δᵒᵈ) hs hf hbot htop | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | ContinuousOn.surjOn_of_tendsto' | If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s`
tends to `Filter.atTop : Filter β` along `Filter.atBot : Filter ↥s` and tends to
`Filter.atBot : Filter β` along `Filter.atTop : Filter ↥s`, then the restriction of `f` to `s` is
surjective. We formulate the conclusion as `Function.surjOn f s Set.univ`. |
Continuous.strictMono_of_inj_boundedOrder [BoundedOrder α] {f : α → δ}
(hf_c : Continuous f) (hf : f ⊥ ≤ f ⊤) (hf_i : Injective f) : StrictMono f := by
intro a b hab
by_contra! h
have H : f b < f a := lt_of_le_of_ne h <| hf_i.ne hab.ne'
by_cases ha : f a ≤ f ⊥
· obtain ⟨u, hu⟩ := intermediate_value_Ioc le_top hf_c.continuousOn ⟨H.trans_le ha, hf⟩
have : u = ⊥ := hf_i hu.2
simp_all
· by_cases hb : f ⊥ < f b
· obtain ⟨u, hu⟩ := intermediate_value_Ioo bot_le hf_c.continuousOn ⟨hb, H⟩
rw [hf_i hu.2] at hu
exact (hab.trans hu.1.2).false
· push_neg at ha hb
replace hb : f b < f ⊥ := lt_of_le_of_ne hb <| hf_i.ne (lt_of_lt_of_le' hab bot_le).ne'
obtain ⟨u, hu⟩ := intermediate_value_Ioo' hab.le hf_c.continuousOn ⟨hb, ha⟩
have : u = ⊥ := hf_i hu.2
simp_all | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | Continuous.strictMono_of_inj_boundedOrder | null |
Continuous.strictAnti_of_inj_boundedOrder [BoundedOrder α] {f : α → δ}
(hf_c : Continuous f) (hf : f ⊤ ≤ f ⊥) (hf_i : Injective f) : StrictAnti f :=
hf_c.strictMono_of_inj_boundedOrder (δ := δᵒᵈ) hf hf_i | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | Continuous.strictAnti_of_inj_boundedOrder | null |
Continuous.strictMono_of_inj_boundedOrder' [BoundedOrder α] {f : α → δ}
(hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f :=
(le_total (f ⊥) (f ⊤)).imp
(hf_c.strictMono_of_inj_boundedOrder · hf_i)
(hf_c.strictAnti_of_inj_boundedOrder · hf_i) | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | Continuous.strictMono_of_inj_boundedOrder' | null |
Continuous.strictMonoOn_of_inj_rigidity {f : α → δ}
(hf_c : Continuous f) (hf_i : Injective f) {a b : α} (hab : a < b)
(hf_mono : StrictMonoOn f (Icc a b)) : StrictMono f := by
intro x y hxy
let s := min a x
let t := max b y
have hsa : s ≤ a := min_le_left a x
have hbt : b ≤ t := le_max_left b y
have hf_mono_st : StrictMonoOn f (Icc s t) ∨ StrictAntiOn f (Icc s t) := by
have : Fact (s ≤ t) := ⟨hsa.trans <| hbt.trans' hab.le⟩
have := Continuous.strictMono_of_inj_boundedOrder' (f := Set.restrict (Icc s t) f)
hf_c.continuousOn.restrict hf_i.injOn.injective
exact this.imp strictMono_restrict.mp strictAntiOn_iff_strictAnti.mpr
have (h : StrictAntiOn f (Icc s t)) : False := by
have : Icc a b ⊆ Icc s t := Icc_subset_Icc hsa hbt
replace : StrictAntiOn f (Icc a b) := StrictAntiOn.mono h this
replace : IsAntichain (· ≤ ·) (Icc a b) :=
IsAntichain.of_strictMonoOn_antitoneOn hf_mono this.antitoneOn
exact this.not_lt (left_mem_Icc.mpr (le_of_lt hab)) (right_mem_Icc.mpr (le_of_lt hab)) hab
replace hf_mono_st : StrictMonoOn f (Icc s t) := hf_mono_st.resolve_right this
have hsx : s ≤ x := min_le_right a x
have hyt : y ≤ t := le_max_right b y
replace : Icc x y ⊆ Icc s t := Icc_subset_Icc hsx hyt
replace : StrictMonoOn f (Icc x y) := StrictMonoOn.mono hf_mono_st this
exact this (left_mem_Icc.mpr (le_of_lt hxy)) (right_mem_Icc.mpr (le_of_lt hxy)) hxy | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | Continuous.strictMonoOn_of_inj_rigidity | Suppose `α` is equipped with a conditionally complete linear dense order and `f : α → δ` is
continuous and injective. Then `f` is strictly monotone (increasing) if
it is strictly monotone (increasing) on some closed interval `[a, b]`. |
ContinuousOn.strictMonoOn_of_injOn_Icc {a b : α} {f : α → δ}
(hab : a ≤ b) (hfab : f a ≤ f b)
(hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :
StrictMonoOn f (Icc a b) := by
have : Fact (a ≤ b) := ⟨hab⟩
refine StrictMono.of_restrict ?_
set g : Icc a b → δ := Set.restrict (Icc a b) f
have hgab : g ⊥ ≤ g ⊤ := by aesop
exact Continuous.strictMono_of_inj_boundedOrder (f := g) hf_c.restrict hgab hf_i.injective | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | ContinuousOn.strictMonoOn_of_injOn_Icc | Suppose `f : [a, b] → δ` is
continuous and injective. Then `f` is strictly monotone (increasing) if `f(a) ≤ f(b)`. |
ContinuousOn.strictAntiOn_of_injOn_Icc {a b : α} {f : α → δ}
(hab : a ≤ b) (hfab : f b ≤ f a)
(hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :
StrictAntiOn f (Icc a b) := ContinuousOn.strictMonoOn_of_injOn_Icc (δ := δᵒᵈ) hab hfab hf_c hf_i | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | ContinuousOn.strictAntiOn_of_injOn_Icc | Suppose `f : [a, b] → δ` is
continuous and injective. Then `f` is strictly antitone (decreasing) if `f(b) ≤ f(a)`. |
ContinuousOn.strictMonoOn_of_injOn_Icc' {a b : α} {f : α → δ} (hab : a ≤ b)
(hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :
StrictMonoOn f (Icc a b) ∨ StrictAntiOn f (Icc a b) :=
(le_total (f a) (f b)).imp
(ContinuousOn.strictMonoOn_of_injOn_Icc hab · hf_c hf_i)
(ContinuousOn.strictAntiOn_of_injOn_Icc hab · hf_c hf_i) | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | ContinuousOn.strictMonoOn_of_injOn_Icc' | Suppose `f : [a, b] → δ` is continuous and injective. Then `f` is strictly monotone
or antitone (increasing or decreasing). |
Continuous.strictMono_of_inj {f : α → δ}
(hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f := by
have H {c d : α} (hcd : c < d) : StrictMono f ∨ StrictAnti f :=
(hf_c.continuousOn.strictMonoOn_of_injOn_Icc' hcd.le hf_i.injOn).imp
(hf_c.strictMonoOn_of_inj_rigidity hf_i hcd)
(hf_c.strictMonoOn_of_inj_rigidity (δ := δᵒᵈ) hf_i hcd)
cases subsingleton_or_nontrivial α with
| inl h => exact Or.inl <| Subsingleton.strictMono f
| inr h =>
obtain ⟨a, b, hab⟩ := exists_pair_lt α
exact H hab | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | Continuous.strictMono_of_inj | Suppose `α` is equipped with a conditionally complete linear dense order and `f : α → δ` is
continuous and injective. Then `f` is strictly monotone or antitone (increasing or decreasing). |
ContinuousOn.strictMonoOn_of_injOn_Ioo {a b : α} {f : α → δ} (hab : a < b)
(hf_c : ContinuousOn f (Ioo a b)) (hf_i : InjOn f (Ioo a b)) :
StrictMonoOn f (Ioo a b) ∨ StrictAntiOn f (Ioo a b) := by
haveI : Inhabited (Ioo a b) := Classical.inhabited_of_nonempty (nonempty_Ioo_subtype hab)
let g : Ioo a b → δ := Set.restrict (Ioo a b) f
have : StrictMono g ∨ StrictAnti g :=
Continuous.strictMono_of_inj hf_c.restrict hf_i.injective
exact this.imp strictMono_restrict.mp strictAntiOn_iff_strictAnti.mpr | theorem | Topology | [
"Mathlib.Order.Interval.Set.Image",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.Order.Monotone",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Order/IntermediateValue.lean | ContinuousOn.strictMonoOn_of_injOn_Ioo | Every continuous injective `f : (a, b) → δ` is strictly monotone
or antitone (increasing or decreasing). |
isLocallyClosed_Icc [Preorder X] [OrderClosedTopology X] :
IsLocallyClosed (Set.Icc a b) :=
isClosed_Icc.isLocallyClosed | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.LocallyClosed"
] | Mathlib/Topology/Order/IsLocallyClosed.lean | isLocallyClosed_Icc | null |
isLocallyClosed_Ioo [LinearOrder X] [OrderClosedTopology X] :
IsLocallyClosed (Set.Ioo a b) :=
isOpen_Ioo.isLocallyClosed | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.LocallyClosed"
] | Mathlib/Topology/Order/IsLocallyClosed.lean | isLocallyClosed_Ioo | null |
isLocallyClosed_Ici [Preorder X] [ClosedIciTopology X] :
IsLocallyClosed (Set.Ici a) :=
isClosed_Ici.isLocallyClosed | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.LocallyClosed"
] | Mathlib/Topology/Order/IsLocallyClosed.lean | isLocallyClosed_Ici | null |
isLocallyClosed_Iic [Preorder X] [ClosedIicTopology X] :
IsLocallyClosed (Set.Iic a) :=
isClosed_Iic.isLocallyClosed | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.LocallyClosed"
] | Mathlib/Topology/Order/IsLocallyClosed.lean | isLocallyClosed_Iic | null |
isLocallyClosed_Ioi [LinearOrder X] [ClosedIicTopology X] :
IsLocallyClosed (Set.Ioi a) :=
isOpen_Ioi.isLocallyClosed | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.LocallyClosed"
] | Mathlib/Topology/Order/IsLocallyClosed.lean | isLocallyClosed_Ioi | null |
isLocallyClosed_Iio [LinearOrder X] [ClosedIciTopology X] :
IsLocallyClosed (Set.Iio a) :=
isOpen_Iio.isLocallyClosed | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.LocallyClosed"
] | Mathlib/Topology/Order/IsLocallyClosed.lean | isLocallyClosed_Iio | null |
isLocallyClosed_Ioc [LinearOrder X] [ClosedIicTopology X] :
IsLocallyClosed (Set.Ioc a b) := by
rw [← Set.Iic_inter_Ioi]
exact isLocallyClosed_Iic.inter isLocallyClosed_Ioi | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.LocallyClosed"
] | Mathlib/Topology/Order/IsLocallyClosed.lean | isLocallyClosed_Ioc | null |
isLocallyClosed_Ico [LinearOrder X] [ClosedIciTopology X] :
IsLocallyClosed (Set.Ico a b) := by
rw [← Set.Iio_inter_Ici]
exact isLocallyClosed_Iio.inter isLocallyClosed_Ici | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.LocallyClosed"
] | Mathlib/Topology/Order/IsLocallyClosed.lean | isLocallyClosed_Ico | null |
IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :
∃ᶠ x in 𝓝[≤] a, x ∈ s := by
rcases hs with ⟨a', ha'⟩
intro h
rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)
· exact h.self_of_nhdsWithin le_rfl ha'
· rcases (mem_nhdsLE_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩
rcases ha.exists_between hba with ⟨b', hb's, hb'⟩
exact hb hb' hb's | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.frequently_mem | null |
IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :
∃ᶠ x in 𝓝 a, x ∈ s :=
(ha.frequently_mem hs).filter_mono inf_le_left | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.frequently_nhds_mem | null |
IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :
∃ᶠ x in 𝓝[≥] a, x ∈ s :=
IsLUB.frequently_mem (α := αᵒᵈ) ha hs | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.frequently_mem | null |
IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :
∃ᶠ x in 𝓝 a, x ∈ s :=
(ha.frequently_mem hs).filter_mono inf_le_left | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.frequently_nhds_mem | null |
IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s :=
(ha.frequently_nhds_mem hs).mem_closure | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.mem_closure | null |
IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s :=
(ha.frequently_nhds_mem hs).mem_closure | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.mem_closure | null |
IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :
NeBot (𝓝[s] a) :=
mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.nhdsWithin_neBot | null |
IsGLB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :
NeBot (𝓝[s] a) :=
IsLUB.nhdsWithin_neBot (α := αᵒᵈ) ha hs | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.nhdsWithin_neBot | null |
isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)
[NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=
⟨hsa, fun b hb =>
not_lt.1 fun hba =>
have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)
let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this
have : b < b := lt_of_lt_of_le hxb <| hb hxs
lt_irrefl b this⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | isLUB_of_mem_nhds | null |
isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :
IsLUB s a := by
rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf
exact isLUB_of_mem_nhds hsa (mem_principal_self s) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | isLUB_of_mem_closure | null |
isGLB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ lowerBounds s) (hsf : s ∈ f)
[NeBot (f ⊓ 𝓝 a)] :
IsGLB s a :=
isLUB_of_mem_nhds (α := αᵒᵈ) hsa hsf | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | isGLB_of_mem_nhds | null |
isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :
IsGLB s a :=
isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | isGLB_of_mem_closure | null |
IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
{f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by
rintro _ ⟨x, hx, rfl⟩
replace ha := ha.inter_Ici_of_mem hx
haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩
refine ge_of_tendsto (hb.mono_left (nhdsWithin_mono a (inter_subset_left (t := Ici x)))) ?_
exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2 | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.mem_upperBounds_of_tendsto | null |
IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
{s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=
haveI := ha.nhdsWithin_neBot hs
⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>
le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.isLUB_of_tendsto | null |
IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
{f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=
IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.mem_lowerBounds_of_tendsto | null |
IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
{s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :
IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=
IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.isGLB_of_tendsto | null |
IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
{f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=
IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.mem_lowerBounds_of_tendsto | null |
IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
{s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsGLB (f '' s) b :=
IsLUB.isLUB_of_tendsto (γ := γᵒᵈ) hf ha hs hb | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.isGLB_of_tendsto | null |
IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
{f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=
IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.mem_upperBounds_of_tendsto | null |
IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
{s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a) (hs : s.Nonempty)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=
IsGLB.isGLB_of_tendsto (γ := γᵒᵈ) hf ha hs hb | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.isLUB_of_tendsto | null |
IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)
(sc : IsClosed s) : a ∈ s :=
sc.closure_subset <| ha.mem_closure hs
alias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.mem_of_isClosed | null |
IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)
(sc : IsClosed s) : a ∈ s :=
sc.closure_subset <| ha.mem_closure hs
alias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.mem_of_isClosed | null |
isLUB_iff_of_subset_of_subset_closure {α : Type*} [TopologicalSpace α] [Preorder α]
[ClosedIicTopology α] {s t : Set α} (hst : s ⊆ t) (hts : t ⊆ closure s) {x : α} :
IsLUB s x ↔ IsLUB t x :=
isLUB_congr <| (upperBounds_closure (s := s) ▸ upperBounds_mono_set hts).antisymm <|
upperBounds_mono_set hst | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | isLUB_iff_of_subset_of_subset_closure | null |
isGLB_iff_of_subset_of_subset_closure {α : Type*} [TopologicalSpace α] [Preorder α]
[ClosedIciTopology α] {s t : Set α} (hst : s ⊆ t) (hts : t ⊆ closure s) {x : α} :
IsGLB s x ↔ IsGLB t x :=
isLUB_iff_of_subset_of_subset_closure (α := αᵒᵈ) hst hts | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | isGLB_iff_of_subset_of_subset_closure | null |
Dense.isLUB_inter_iff {α : Type*} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α]
{s t : Set α} (hs : Dense s) (ht : IsOpen t) {x : α} :
IsLUB (t ∩ s) x ↔ IsLUB t x :=
isLUB_iff_of_subset_of_subset_closure (by simp) <| hs.open_subset_closure_inter ht | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | Dense.isLUB_inter_iff | null |
Dense.isGLB_inter_iff {α : Type*} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α]
{s t : Set α} (hs : Dense s) (ht : IsOpen t) {x : α} :
IsGLB (t ∩ s) x ↔ IsGLB t x :=
hs.isLUB_inter_iff (α := αᵒᵈ) ht
/-! | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | Dense.isGLB_inter_iff | null |
IsLUB.exists_seq_strictMono_tendsto_of_notMem {t : Set α} {x : α}
[IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (notMem : x ∉ t) (ht : t.Nonempty) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by
obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)
replace hvx := hvx.mono_right nhdsWithin_le_nhds
have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) notMem)
have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')
choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists
refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx',
hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩
· rw [iterate_succ_apply']; exact hvN _
· rw [iterate_succ_apply']; exact hN _
@[deprecated (since := "2025-05-23")]
alias IsLUB.exists_seq_strictMono_tendsto_of_not_mem :=
IsLUB.exists_seq_strictMono_tendsto_of_notMem | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.exists_seq_strictMono_tendsto_of_notMem | null |
IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]
(htx : IsLUB t x) (ht : t.Nonempty) :
∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by
by_cases h : x ∈ t
· exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩
· rcases htx.exists_seq_strictMono_tendsto_of_notMem h ht with ⟨u, hu⟩
exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsLUB.exists_seq_monotone_tendsto | null |
exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]
[DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by
have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim
have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy
rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_notMem hx ht with ⟨u, hu⟩
exact ⟨u, hu.1, hu.2.2.symm⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | exists_seq_strictMono_tendsto' | null |
exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]
(x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by
obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x
rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩
exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | exists_seq_strictMono_tendsto | null |
exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]
[FirstCountableTopology α] (x : α) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) :=
let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x
⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | exists_seq_strictMono_tendsto_nhdsWithin | null |
exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]
[TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)
(hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by
rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩
exact ⟨u, hu.1, hu.2.2⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | exists_seq_tendsto_sSup | null |
Dense.exists_seq_strictMono_tendsto_of_lt [DenselyOrdered α] [FirstCountableTopology α]
{s : Set α} (hs : Dense s) {x y : α} (hy : y < x) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ (Ioo y x ∩ s)) ∧ Tendsto u atTop (𝓝 x) := by
have hnonempty : (Ioo y x ∩ s).Nonempty := by
obtain ⟨z, hyz, hzx⟩ := hs.exists_between hy
exact ⟨z, mem_inter hzx hyz⟩
have hx : IsLUB (Ioo y x ∩ s) x := hs.isLUB_inter_iff isOpen_Ioo |>.mpr <| isLUB_Ioo hy
apply hx.exists_seq_strictMono_tendsto_of_notMem (by simp) hnonempty |>.imp
simp_all | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | Dense.exists_seq_strictMono_tendsto_of_lt | null |
Dense.exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α]
[FirstCountableTopology α] {s : Set α} (hs : Dense s) (x : α) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ (Iio x ∩ s)) ∧ Tendsto u atTop (𝓝 x) := by
obtain ⟨y, hy⟩ := exists_lt x
apply hs.exists_seq_strictMono_tendsto_of_lt (exists_lt x).choose_spec |>.imp
simp_all | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | Dense.exists_seq_strictMono_tendsto | null |
DenseRange.exists_seq_strictMono_tendsto_of_lt {β : Type*} [LinearOrder β]
[DenselyOrdered α] [FirstCountableTopology α] {f : β → α} {x y : α} (hf : DenseRange f)
(hmono : Monotone f) (hlt : y < x) :
∃ u : ℕ → β, StrictMono u ∧ (∀ n, f (u n) ∈ Ioo y x) ∧ Tendsto (f ∘ u) atTop (𝓝 x) := by
rcases Dense.exists_seq_strictMono_tendsto_of_lt hf hlt with ⟨u, hu, huyxf, hlim⟩
have huyx (n : ℕ) : u n ∈ Ioo y x := (huyxf n).1
have huf (n : ℕ) : u n ∈ range f := (huyxf n).2
choose v hv using huf
obtain rfl : f ∘ v = u := funext hv
exact ⟨v, fun a b hlt ↦ hmono.reflect_lt <| hu hlt, huyx, hlim⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | DenseRange.exists_seq_strictMono_tendsto_of_lt | null |
DenseRange.exists_seq_strictMono_tendsto {β : Type*} [LinearOrder β] [DenselyOrdered α]
[NoMinOrder α] [FirstCountableTopology α] {f : β → α} (hf : DenseRange f) (hmono : Monotone f)
(x : α) :
∃ u : ℕ → β, StrictMono u ∧ (∀ n, f (u n) ∈ Iio x) ∧ Tendsto (f ∘ u) atTop (𝓝 x) := by
rcases Dense.exists_seq_strictMono_tendsto hf x with ⟨u, hu, huxf, hlim⟩
have hux (n : ℕ) : u n ∈ Iio x := (huxf n).1
have huf (n : ℕ) : u n ∈ range f := (huxf n).2
choose v hv using huf
obtain rfl : f ∘ v = u := funext hv
exact ⟨v, fun a b hlt ↦ hmono.reflect_lt <| hu hlt, hux, hlim⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | DenseRange.exists_seq_strictMono_tendsto | null |
IsGLB.exists_seq_strictAnti_tendsto_of_notMem {t : Set α} {x : α}
[IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (notMem : x ∉ t) (ht : t.Nonempty) :
∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=
IsLUB.exists_seq_strictMono_tendsto_of_notMem (α := αᵒᵈ) htx notMem ht
@[deprecated (since := "2025-05-23")]
alias IsGLB.exists_seq_strictAnti_tendsto_of_not_mem :=
IsGLB.exists_seq_strictAnti_tendsto_of_notMem | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.exists_seq_strictAnti_tendsto_of_notMem | null |
IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]
(htx : IsGLB t x) (ht : t.Nonempty) :
∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=
IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | IsGLB.exists_seq_antitone_tendsto | null |
exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}
(hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by
simpa using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | exists_seq_strictAnti_tendsto' | null |
exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]
(x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) :=
exists_seq_strictMono_tendsto (α := αᵒᵈ) x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | exists_seq_strictAnti_tendsto | null |
exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]
[FirstCountableTopology α] (x : α) :
∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) :=
exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | exists_seq_strictAnti_tendsto_nhdsWithin | null |
exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]
{x y : α} (h : x < y) :
∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧
(∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by
rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩
rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩
exact
⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,
fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩ | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | exists_seq_strictAnti_strictMono_tendsto | null |
exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]
[TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)
(hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S :=
exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS' | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | exists_seq_tendsto_sInf | null |
Dense.exists_seq_strictAnti_tendsto_of_lt [DenselyOrdered α] [FirstCountableTopology α]
{s : Set α} (hs : Dense s) {x y : α} (hy : x < y) :
∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ (Ioo x y ∩ s)) ∧ Tendsto u atTop (𝓝 x) := by
simpa using hs.exists_seq_strictMono_tendsto_of_lt (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | Dense.exists_seq_strictAnti_tendsto_of_lt | null |
Dense.exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α]
[FirstCountableTopology α] {s : Set α} (hs : Dense s) (x : α) :
∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ (Ioi x ∩ s)) ∧ Tendsto u atTop (𝓝 x) :=
hs.exists_seq_strictMono_tendsto (α := αᵒᵈ) x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | Dense.exists_seq_strictAnti_tendsto | null |
DenseRange.exists_seq_strictAnti_tendsto_of_lt {β : Type*} [LinearOrder β]
[DenselyOrdered α] [FirstCountableTopology α] {f : β → α} {x y : α} (hf : DenseRange f)
(hmono : Monotone f) (hlt : x < y) :
∃ u : ℕ → β, StrictAnti u ∧ (∀ n, f (u n) ∈ Ioo x y) ∧ Tendsto (f ∘ u) atTop (𝓝 x) := by
simpa using hf.exists_seq_strictMono_tendsto_of_lt (α := αᵒᵈ) (β := βᵒᵈ) hmono.dual
(OrderDual.toDual_lt_toDual.2 hlt) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | DenseRange.exists_seq_strictAnti_tendsto_of_lt | null |
DenseRange.exists_seq_strictAnti_tendsto {β : Type*} [LinearOrder β] [DenselyOrdered α]
[NoMaxOrder α] [FirstCountableTopology α] {f : β → α} (hf : DenseRange f) (hmono : Monotone f)
(x : α) :
∃ u : ℕ → β, StrictAnti u ∧ (∀ n, f (u n) ∈ Ioi x) ∧ Tendsto (f ∘ u) atTop (𝓝 x) :=
hf.exists_seq_strictMono_tendsto (α := αᵒᵈ) (β := βᵒᵈ) hmono.dual x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRightNhds"
] | Mathlib/Topology/Order/IsLUB.lean | DenseRange.exists_seq_strictAnti_tendsto | null |
IsNormal.continuous {f : α → β} (hf : IsNormal f) : Continuous f := by
rw [OrderTopology.continuous_iff]
refine fun b ↦ ⟨?_, ((isLowerSet_Iio b).preimage hf.strictMono.monotone).isOpen⟩
rw [← isClosed_compl_iff, ← Set.preimage_compl, Set.compl_Ioi]
obtain ha | ⟨a, ha⟩ := ((isLowerSet_Iic b).preimage hf.strictMono.monotone).eq_univ_or_Iio
· exact ha ▸ isClosed_univ
· obtain h | h := (f ⁻¹' Iic b).eq_empty_or_nonempty
· exact h ▸ isClosed_empty
· have : Nonempty α := ⟨a⟩
have : Nonempty β := ⟨b⟩
rw [hf.preimage_Iic h (ha ▸ bddAbove_Iio)]
exact isClosed_Iic | theorem | Topology | [
"Mathlib.Order.IsNormal",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/IsNormal.lean | IsNormal.continuous | null |
isNormal_iff_strictMono_and_continuous {f : α → β} :
IsNormal f ↔ StrictMono f ∧ Continuous f where
mp hf := ⟨hf.strictMono, hf.continuous⟩
mpr := by
rintro ⟨hs, hc⟩
refine ⟨hs, fun {a} ha ↦ (isLUB_of_mem_closure ?_ ?_).2⟩
· rintro _ ⟨b, hb, rfl⟩
exact (hs hb).le
· apply image_closure_subset_closure_image hc (mem_image_of_mem ..)
exact ha.isLUB_Iio.mem_closure (Iio_nonempty.2 ha.1) | theorem | Topology | [
"Mathlib.Order.IsNormal",
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/IsNormal.lean | isNormal_iff_strictMono_and_continuous | A normal function between well-orders is equivalent to a strictly monotone,
continuous function. |
ContinuousInf (L : Type*) [TopologicalSpace L] [Min L] : Prop where
/-- The infimum is continuous -/
continuous_inf : Continuous fun p : L × L => p.1 ⊓ p.2 | class | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousInf | 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. |
ContinuousSup (L : Type*) [TopologicalSpace L] [Max L] : Prop where
/-- The supremum is continuous -/
continuous_sup : Continuous fun p : L × L => p.1 ⊔ p.2 | class | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | ContinuousSup | 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. |
TopologicalLattice (L : Type*) [TopologicalSpace L] [Lattice L] : Prop
extends ContinuousInf L, ContinuousSup L | class | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | TopologicalLattice | 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*. |
@[continuity]
continuous_inf [Min L] [ContinuousInf L] : Continuous fun p : L × L => p.1 ⊓ p.2 :=
ContinuousInf.continuous_inf
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | continuous_inf | null |
Continuous.inf [Min L] [ContinuousInf L] {f g : X → L} (hf : Continuous f)
(hg : Continuous g) : Continuous fun x => f x ⊓ g x :=
continuous_inf.comp (hf.prodMk hg :)
@[continuity] | theorem | Topology | [
"Mathlib.Topology.Constructions",
"Mathlib.Topology.Order.OrderClosed"
] | Mathlib/Topology/Order/Lattice.lean | Continuous.inf | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.