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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comap_coe_nhdsGT_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) :
comap ((↑) : s → α) (𝓝[>] a) = atBot := by
apply comap_coe_nhdsLT_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)
simp only [OrderDual.exists, Ioo_toDual]
exact hs | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_nhdsGT_of_Ioo_subset | null |
map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b := by
rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)
· have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩
rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]
· rw [← comap_coe_nhdsLT_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]
rw [Subtype.range_val]
exact (mem_nhdsLT_iff_exists_Ioo_subset' ha).2 (hs a ha) | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_atTop_of_Ioo_subset | null |
map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :
map ((↑) : s → α) atBot = 𝓝[>] a := by
refine (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha)
fun b' hb' => ?_ :)
simpa using hs b' hb' | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_atBot_of_Ioo_subset | null |
comap_coe_Ioo_nhdsLT (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop :=
comap_coe_nhdsLT_of_Ioo_subset Ioo_subset_Iio_self fun h => ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩ | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_Ioo_nhdsLT | The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at
the right endpoint in the ambient order. |
comap_coe_Ioo_nhdsGT (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot :=
comap_coe_nhdsGT_of_Ioo_subset Ioo_subset_Ioi_self fun h => ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩ | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_Ioo_nhdsGT | The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at
the left endpoint in the ambient order. |
comap_coe_Ioi_nhdsGT (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot :=
comap_coe_nhdsGT_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩ | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_Ioi_nhdsGT | null |
comap_coe_Iio_nhdsLT (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop :=
comap_coe_Ioi_nhdsGT (α := αᵒᵈ) a
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_Iio_nhdsLT | null |
map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b :=
map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_Ioo_atTop | null |
map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a :=
map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_Ioo_atBot | null |
map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a :=
map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_Ioi_atBot | null |
map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a :=
map_coe_Ioi_atBot (α := αᵒᵈ) _
variable {l : Filter β} {f : α → β}
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | map_coe_Iio_atTop | null |
tendsto_comp_coe_Ioo_atTop (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by
rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_comp_coe_Ioo_atTop | null |
tendsto_comp_coe_Ioo_atBot (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by
rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_comp_coe_Ioo_atBot | null |
tendsto_comp_coe_Ioi_atBot :
Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by
rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_comp_coe_Ioi_atBot | null |
tendsto_comp_coe_Iio_atTop :
Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by
rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_comp_coe_Iio_atTop | null |
tendsto_Ioo_atTop {f : β → Ioo a b} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by
rw [← comap_coe_Ioo_nhdsLT, tendsto_comap_iff]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_Ioo_atTop | null |
tendsto_Ioo_atBot {f : β → Ioo a b} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by
rw [← comap_coe_Ioo_nhdsGT, tendsto_comap_iff]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_Ioo_atBot | null |
tendsto_Ioi_atBot {f : β → Ioi a} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by
rw [← comap_coe_Ioi_nhdsGT, tendsto_comap_iff]; rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_Ioi_atBot | null |
tendsto_Iio_atTop {f : β → Iio a} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by
rw [← comap_coe_Iio_nhdsLT, tendsto_comap_iff]; rfl | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | tendsto_Iio_atTop | null |
Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s]
(hs : Dense s) :
∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t := by
rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩
refine ⟨t \ ({ x | IsBot x } ∪ { x | IsTop x }), ?_, ?_, ?_, fun x hx => ?_, fun x hx => ?_⟩
· exact diff_subset.trans hts
· exact htc.mono diff_subset
· exact htd.diff_finite ((subsingleton_isBot α).finite.union (subsingleton_isTop α).finite)
· simp [hx]
· simp [hx]
variable (α) in | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | Dense.exists_countable_dense_subset_no_bot_top | Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. |
exists_countable_dense_no_bot_top [SeparableSpace α] [Nontrivial α] :
∃ s : Set α, s.Countable ∧ Dense s ∧ (∀ x, IsBot x → x ∉ s) ∧ ∀ x, IsTop x → x ∉ s := by
simpa using dense_univ.exists_countable_dense_subset_no_bot_top | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | exists_countable_dense_no_bot_top | If `α` is a nontrivial separable dense linear order, then there exists a
countable dense set `s : Set α` that contains neither top nor bottom elements of `α`.
For a dense set containing both bot and top elements, see
`exists_countable_dense_bot_top`. |
@[simp]
isClosed_Ico_iff {a b : α} : IsClosed (Set.Ico a b) ↔ b ≤ a := by
refine ⟨fun h => le_of_not_gt fun hab => ?_, by simp_all⟩
have := h.closure_eq
rw [closure_Ico hab.ne, Icc_eq_Ico_same_iff] at this
exact this hab.le | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | isClosed_Ico_iff | `Set.Ico a b` is only closed if it is empty. |
@[simp]
isClosed_Ioc_iff {a b : α} : IsClosed (Set.Ioc a b) ↔ b ≤ a := by
refine ⟨fun h => le_of_not_gt fun hab => ?_, by simp_all⟩
have := h.closure_eq
rw [closure_Ioc hab.ne, Icc_eq_Ioc_same_iff] at this
exact this hab.le | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | isClosed_Ioc_iff | `Set.Ioc a b` is only closed if it is empty. |
@[simp]
isClosed_Ioo_iff {a b : α} : IsClosed (Set.Ioo a b) ↔ b ≤ a := by
refine ⟨fun h => le_of_not_gt fun hab => ?_, by simp_all⟩
have := h.closure_eq
rw [closure_Ioo hab.ne, Icc_eq_Ioo_same_iff] at this
exact this hab.le | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | isClosed_Ioo_iff | `Set.Ioo a b` is only closed if it is empty. |
continuousOn_Icc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la lb : β}
(hab : a ≠ b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la))
(hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : ContinuousOn (extendFrom (Ioo a b) f) (Icc a b) := by
apply continuousOn_extendFrom
· rw [closure_Ioo hab]
· intro x x_in
rcases eq_endpoints_or_mem_Ioo_of_mem_Icc x_in with (rfl | rfl | h)
· exact ⟨la, ha.mono_left <| nhdsWithin_mono _ Ioo_subset_Ioi_self⟩
· exact ⟨lb, hb.mono_left <| nhdsWithin_mono _ Ioo_subset_Iio_self⟩
· exact ⟨f x, hf x h⟩ | theorem | Topology | [
"Mathlib.Topology.ExtendFrom",
"Mathlib.Topology.Order.DenselyOrdered"
] | Mathlib/Topology/Order/ExtendFrom.lean | continuousOn_Icc_extendFrom_Ioo | null |
eq_lim_at_left_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {la : β} (hab : a < b)
(ha : Tendsto f (𝓝[>] a) (𝓝 la)) : extendFrom (Ioo a b) f a = la := by
apply extendFrom_eq
· rw [closure_Ioo hab.ne]
simp only [le_of_lt hab, left_mem_Icc]
· simpa [hab] | theorem | Topology | [
"Mathlib.Topology.ExtendFrom",
"Mathlib.Topology.Order.DenselyOrdered"
] | Mathlib/Topology/Order/ExtendFrom.lean | eq_lim_at_left_extendFrom_Ioo | null |
eq_lim_at_right_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {lb : β} (hab : a < b)
(hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : extendFrom (Ioo a b) f b = lb := by
apply extendFrom_eq
· rw [closure_Ioo hab.ne]
simp only [le_of_lt hab, right_mem_Icc]
· simpa [hab] | theorem | Topology | [
"Mathlib.Topology.ExtendFrom",
"Mathlib.Topology.Order.DenselyOrdered"
] | Mathlib/Topology/Order/ExtendFrom.lean | eq_lim_at_right_extendFrom_Ioo | null |
continuousOn_Ico_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la : β}
(hab : a < b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la)) :
ContinuousOn (extendFrom (Ioo a b) f) (Ico a b) := by
apply continuousOn_extendFrom
· rw [closure_Ioo hab.ne]
exact Ico_subset_Icc_self
· intro x x_in
rcases eq_left_or_mem_Ioo_of_mem_Ico x_in with (rfl | h)
· use la
simpa [hab]
· exact ⟨f x, hf x h⟩ | theorem | Topology | [
"Mathlib.Topology.ExtendFrom",
"Mathlib.Topology.Order.DenselyOrdered"
] | Mathlib/Topology/Order/ExtendFrom.lean | continuousOn_Ico_extendFrom_Ioo | null |
continuousOn_Ioc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {lb : β}
(hab : a < b) (hf : ContinuousOn f (Ioo a b)) (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) :
ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b) := by
have := @continuousOn_Ico_extendFrom_Ioo αᵒᵈ _ _ _ _ _ _ _ f _ _ lb hab
erw [Ico_toDual, Ioi_toDual, Ioo_toDual] at this
exact this hf hb | theorem | Topology | [
"Mathlib.Topology.ExtendFrom",
"Mathlib.Topology.Order.DenselyOrdered"
] | Mathlib/Topology/Order/ExtendFrom.lean | continuousOn_Ioc_extendFrom_Ioo | null |
protected IsMaxOn.closure (h : IsMaxOn f s a) (hc : ContinuousOn f (closure s)) :
IsMaxOn f (closure s) a := fun x hx =>
ContinuousWithinAt.closure_le hx ((hc x hx).mono subset_closure) continuousWithinAt_const h | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsMaxOn.closure | null |
protected IsMinOn.closure (h : IsMinOn f s a) (hc : ContinuousOn f (closure s)) :
IsMinOn f (closure s) a :=
h.dual.closure hc | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsMinOn.closure | null |
protected IsExtrOn.closure (h : IsExtrOn f s a) (hc : ContinuousOn f (closure s)) :
IsExtrOn f (closure s) a :=
h.elim (fun h => Or.inl <| h.closure hc) fun h => Or.inr <| h.closure hc | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsExtrOn.closure | null |
protected IsLocalMaxOn.closure (h : IsLocalMaxOn f s a) (hc : ContinuousOn f (closure s)) :
IsLocalMaxOn f (closure s) a := by
rcases mem_nhdsWithin.1 h with ⟨U, Uo, aU, hU⟩
refine mem_nhdsWithin.2 ⟨U, Uo, aU, ?_⟩
rintro x ⟨hxU, hxs⟩
refine ContinuousWithinAt.closure_le ?_ ?_ continuousWithinAt_const hU
· rwa [mem_closure_iff_nhdsWithin_neBot, nhdsWithin_inter_of_mem, ←
mem_closure_iff_nhdsWithin_neBot]
exact nhdsWithin_le_nhds (Uo.mem_nhds hxU)
· exact (hc _ hxs).mono (inter_subset_right.trans subset_closure) | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsLocalMaxOn.closure | null |
protected IsLocalMinOn.closure (h : IsLocalMinOn f s a) (hc : ContinuousOn f (closure s)) :
IsLocalMinOn f (closure s) a :=
IsLocalMaxOn.closure h.dual hc | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsLocalMinOn.closure | null |
protected IsLocalExtrOn.closure (h : IsLocalExtrOn f s a)
(hc : ContinuousOn f (closure s)) : IsLocalExtrOn f (closure s) a :=
h.elim (fun h => Or.inl <| h.closure hc) fun h => Or.inr <| h.closure hc | theorem | Topology | [
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/ExtrClosure.lean | IsLocalExtrOn.closure | null |
protected tendsto_nhds_atTop [NoMaxOrder X] : Tendsto 𝓝 (atTop : Filter X) (𝓝 atTop) :=
Filter.tendsto_nhds_atTop_iff.2 fun x => (eventually_gt_atTop x).mono fun _ => le_mem_nhds | theorem | Topology | [
"Mathlib.Topology.Filter",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/Filter.lean | tendsto_nhds_atTop | null |
protected tendsto_nhds_atBot [NoMinOrder X] : Tendsto 𝓝 (atBot : Filter X) (𝓝 atBot) :=
@Filter.tendsto_nhds_atTop Xᵒᵈ _ _ _ _ | theorem | Topology | [
"Mathlib.Topology.Filter",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/Filter.lean | tendsto_nhds_atBot | null |
Tendsto.nhds_atTop [NoMaxOrder X] {f : α → X} {l : Filter α} (h : Tendsto f l atTop) :
Tendsto (𝓝 ∘ f) l (𝓝 atTop) :=
Filter.tendsto_nhds_atTop.comp h | theorem | Topology | [
"Mathlib.Topology.Filter",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/Filter.lean | Tendsto.nhds_atTop | null |
Tendsto.nhds_atBot [NoMinOrder X] {f : α → X} {l : Filter α} (h : Tendsto f l atBot) :
Tendsto (𝓝 ∘ f) l (𝓝 atBot) :=
@Tendsto.nhds_atTop α Xᵒᵈ _ _ _ _ _ _ h | theorem | Topology | [
"Mathlib.Topology.Filter",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/Filter.lean | Tendsto.nhds_atBot | null |
hull (T : Set α) (a : α) := T ↓∩ Ici a
variable {T : Set α}
/- The set of relative-closed sets of the form `hull T a` for some `a` in `α` is closed under
pairwise union. -/ | abbrev | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull | For `a` of type `α` the set of element of `T` which dominate `a` is the `hull` of `a` in `T`. |
hull_inf (hT : ∀ p ∈ T, InfPrime p) (a b : α) :
hull T (a ⊓ b) = hull T a ∪ hull T b := by
ext p
constructor <;> intro h
· exact (hT p p.2).2 h
· rcases h with (h1 | h3)
· exact inf_le_of_left_le h1
· exact inf_le_of_right_le h3
variable [OrderTop α]
/- Every relative-closed set of the form `T ↓∩ (↑(upperClosure F))` for `F` finite is a
relative-closed set of the form `hull T a` where `a = ⨅ F`. -/
open Finset in | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull_inf | null |
hull_finsetInf (hT : ∀ p ∈ T, InfPrime p) (F : Finset α) :
hull T (inf F id) = T ↓∩ upperClosure F.toSet := by
rw [coe_upperClosure]
induction F using Finset.cons_induction with
| empty =>
simp only [coe_empty, mem_empty_iff_false, iUnion_of_empty, iUnion_empty, Set.preimage_empty,
inf_empty]
by_contra hf
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hf
obtain ⟨x, hx⟩ := hf
exact (hT x (Subtype.coe_prop x)).1 (isMax_iff_eq_top.mpr (eq_top_iff.mpr hx))
| cons a F' _ I4 => simp [hull_inf hT, I4]
/- Every relative-open set of the form `T ↓∩ (↑(upperClosure F))ᶜ` for `F` finite is a relative-open
set of the form `(hull T a)ᶜ` where `a = ⨅ F`. -/
open Finset in | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull_finsetInf | null |
preimage_upperClosure_compl_finset (hT : ∀ p ∈ T, InfPrime p) (F : Finset α) :
T ↓∩ (upperClosure F.toSet)ᶜ = (hull T (inf F id))ᶜ := by
rw [Set.preimage_compl, (hull_finsetInf hT)]
variable [TopologicalSpace α] [IsLower α]
/-
The relative-open sets of the form `(hull T a)ᶜ` for `a` in `α` form a basis for the relative
Lower topology.
-/ | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | preimage_upperClosure_compl_finset | null |
isTopologicalBasis_relativeLower (hT : ∀ p ∈ T, InfPrime p) :
IsTopologicalBasis { S : Set T | ∃ (a : α), (hull T a)ᶜ = S } := by
convert isTopologicalBasis_subtype Topology.IsLower.isTopologicalBasis T
ext R
simp only [preimage_compl, mem_setOf_eq, IsLower.lowerBasis, mem_image, exists_exists_and_eq_and]
constructor <;> intro ha
· obtain ⟨a, ha'⟩ := ha
use {a}
rw [← (Function.Injective.preimage_image Subtype.val_injective R), ← ha']
simp only [finite_singleton, upperClosure_singleton, UpperSet.coe_Ici, image_val_compl,
Subtype.image_preimage_coe, diff_self_inter, preimage_diff, Subtype.coe_preimage_self,
true_and]
exact compl_eq_univ_diff (Subtype.val ⁻¹' Ici a)
· obtain ⟨F, hF⟩ := ha
lift F to Finset α using hF.1
use Finset.inf F id
ext
simp [hull_finsetInf hT, ← hF.2] | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | isTopologicalBasis_relativeLower | null |
hull_iSup {ι : Sort v} (s : ι → α) : hull T (iSup s) = ⋂ i, hull T (s i) := by aesop | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull_iSup | null |
hull_sSup (S : Set α) : hull T (sSup S) = ⋂₀ { hull T a | a ∈ S } := by aesop
/- When `α` is complete, a set is Lower topology relative-open if and only if it is of the form
`(hull T a)ᶜ` for some `a` in `α`.-/ | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull_sSup | null |
isOpen_iff [TopologicalSpace α] [IsLower α] (hT : ∀ p ∈ T, InfPrime p)
(S : Set T) : IsOpen S ↔ ∃ (a : α), S = (hull T a)ᶜ := by
constructor <;> intro h
· let R := {a : α | (hull T a)ᶜ ⊆ S}
use sSup R
rw [IsTopologicalBasis.open_eq_sUnion' (isTopologicalBasis_relativeLower hT) h]
aesop
· obtain ⟨a, ha⟩ := h
exact ⟨(Ici a)ᶜ, isClosed_Ici.isOpen_compl, ha.symm⟩
/- When `α` is complete, a set is closed in the relative lower topology if and only if it is of the
form `hull T a` for some `a` in `α`.-/ | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | isOpen_iff | null |
isClosed_iff [TopologicalSpace α] [IsLower α] (hT : ∀ p ∈ T, InfPrime p)
{S : Set T} : IsClosed S ↔ ∃ (a : α), S = hull T a := by
simp only [← isOpen_compl_iff, isOpen_iff hT, compl_inj_iff] | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | isClosed_iff | null |
kernel (S : Set T) := sInf (Subtype.val '' S)
/- The pair of maps `kernel` and `hull` form an antitone Galois connection between the
subsets of `T` and `α`. -/
open OrderDual in | abbrev | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | kernel | For a subset `S` of `T`, `kernel S` is the infimum of `S` (considered as a set of `α`) |
gc : GaloisConnection (α := Set T) (β := αᵒᵈ)
(fun S => toDual (kernel S)) (fun a => hull T (ofDual a)) := fun S a => by
simp only [toDual_sInf, sSup_le_iff, mem_preimage, mem_image, Subtype.exists, exists_and_right,
exists_eq_right, ← ofDual_le_ofDual, forall_exists_index, OrderDual.forall, ofDual_toDual]
exact ⟨fun h b hbS => h _ (Subtype.coe_prop b) hbS, fun h b _ hbS => h hbS⟩ | theorem | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | gc | null |
gc_closureOperator (S : Set T) : gc.closureOperator S = hull T (kernel S) := by
simp only [toDual_sInf, GaloisConnection.closureOperator_apply, ofDual_sSup]
rw [← preimage_comp, ← OrderDual.toDual_symm_eq, Equiv.symm_comp_self, preimage_id_eq, id_eq]
variable (T) | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | gc_closureOperator | null |
OrderGenerates := ∀ (a : α), ∃ (S : Set T), a = kernel S
variable {T} | def | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | OrderGenerates | `T` order generates `α` if, for every `a` in `α`, there exists a subset of `T` such that `a` is
the `kernel` of `S`. |
gi (hG : OrderGenerates T) : GaloisInsertion (α := Set T) (β := αᵒᵈ)
(OrderDual.toDual ∘ kernel)
(hull T ∘ OrderDual.ofDual) :=
gc.toGaloisInsertion fun a ↦ by
rw [OrderDual.le_toDual]
obtain ⟨S, hS⟩ := hG a
exact le_of_le_of_eq (sInf_le_sInf (image_val_mono (fun c hcS => mem_preimage.mpr (mem_Ici.mpr
(by rw [hS]; exact CompleteSemilatticeInf.sInf_le _ _ (mem_image_of_mem Subtype.val hcS))))))
hS.symm | def | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | gi | When `T` is order generating, the `kernel` and the `hull` form a Galois insertion |
kernel_hull (hG : OrderGenerates T) (a : α) : kernel (hull T a) = a := by
conv_rhs => rw [← OrderDual.ofDual_toDual a, ← (gi hG).l_u_eq a]
rfl | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | kernel_hull | null |
hull_kernel_of_isClosed [TopologicalSpace α] [IsLower α]
(hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T) {C : Set T} (h : IsClosed C) :
hull T (kernel C) = C := by
obtain ⟨a, ha⟩ := (isClosed_iff hT).mp h
rw [ha, kernel_hull hG] | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | hull_kernel_of_isClosed | null |
closedsGC_closureOperator [TopologicalSpace α] [IsLower α]
(hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T) (S : Set T) :
(TopologicalSpace.Closeds.gc (α := T)).closureOperator S = hull T (kernel S) := by
simp only [GaloisConnection.closureOperator_apply, Closeds.coe_closure, closure, le_antisymm_iff]
constructor
· exact fun ⦃a⦄ a ↦ a (hull T (kernel S)) ⟨(isClosed_iff hT).mpr ⟨kernel S, rfl⟩,
image_subset_iff.mp (fun _ hbS => CompleteSemilatticeInf.sInf_le _ _ hbS)⟩
· simp_rw [le_eq_subset, subset_sInter_iff]
intro R hR
rw [← (hull_kernel_of_isClosed hT hG hR.1), ← gc_closureOperator]
exact ClosureOperator.monotone _ hR.2 | lemma | Topology | [
"Mathlib.Data.Set.Subset",
"Mathlib.Order.Irreducible",
"Mathlib.Topology.Order.LowerUpperTopology",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Order/HullKernel.lean | closedsGC_closureOperator | null |
intermediate_value_univ₂ [PreconnectedSpace X] {a b : X} {f g : X → α} (hf : Continuous f)
(hg : Continuous g) (ha : f a ≤ g a) (hb : g b ≤ f b) : ∃ x, f x = g x := by
obtain ⟨x, _, hfg, hgf⟩ : (univ ∩ { x | f x ≤ g x ∧ g x ≤ f x }).Nonempty :=
isPreconnected_closed_iff.1 PreconnectedSpace.isPreconnected_univ _ _ (isClosed_le hf hg)
(isClosed_le hg hf) (fun _ _ => le_total _ _) ⟨a, trivial, ha⟩ ⟨b, trivial, hb⟩
exact ⟨x, le_antisymm hfg hgf⟩ | 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_univ₂ | Intermediate value theorem for two functions: if `f` and `g` are two continuous functions
on a preconnected space and `f a ≤ g a` and `g b ≤ f b`, then for some `x` we have `f x = g x`. |
intermediate_value_univ₂_eventually₁ [PreconnectedSpace X] {a : X} {l : Filter X} [NeBot l]
{f g : X → α} (hf : Continuous f) (hg : Continuous g) (ha : f a ≤ g a) (he : g ≤ᶠ[l] f) :
∃ x, f x = g x :=
let ⟨_, h⟩ := he.exists; intermediate_value_univ₂ hf hg ha h | 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_univ₂_eventually₁ | null |
intermediate_value_univ₂_eventually₂ [PreconnectedSpace X] {l₁ l₂ : Filter X} [NeBot l₁]
[NeBot l₂] {f g : X → α} (hf : Continuous f) (hg : Continuous g) (he₁ : f ≤ᶠ[l₁] g)
(he₂ : g ≤ᶠ[l₂] f) : ∃ x, f x = g x :=
let ⟨_, h₁⟩ := he₁.exists
let ⟨_, h₂⟩ := he₂.exists
intermediate_value_univ₂ hf hg h₁ h₂ | 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_univ₂_eventually₂ | null |
IsPreconnected.intermediate_value₂ {s : Set X} (hs : IsPreconnected s) {a b : X}
(ha : a ∈ s) (hb : b ∈ s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
(ha' : f a ≤ g a) (hb' : g b ≤ f b) : ∃ x ∈ s, f x = g x :=
let ⟨x, hx⟩ :=
@intermediate_value_univ₂ s α _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ ⟨b, hb⟩ _ _
(continuousOn_iff_continuous_restrict.1 hf) (continuousOn_iff_continuous_restrict.1 hg) ha'
hb'
⟨x, x.2, hx⟩ | 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.intermediate_value₂ | Intermediate value theorem for two functions: if `f` and `g` are two functions continuous
on a preconnected set `s` and for some `a b ∈ s` we have `f a ≤ g a` and `g b ≤ f b`,
then for some `x ∈ s` we have `f x = g x`. |
IsPreconnected.intermediate_value₂_eventually₁ {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s)
(hg : ContinuousOn g s) (ha' : f a ≤ g a) (he : g ≤ᶠ[l] f) : ∃ x ∈ s, f x = g x := by
rw [continuousOn_iff_continuous_restrict] at hf hg
obtain ⟨b, h⟩ :=
@intermediate_value_univ₂_eventually₁ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ _
(comap_coe_neBot_of_le_principal hl) _ _ hf hg ha' (he.comap _)
exact ⟨b, b.prop, h⟩ | 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.intermediate_value₂_eventually₁ | null |
IsPreconnected.intermediate_value₂_eventually₂ {s : Set X} (hs : IsPreconnected s)
{l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f g : X → α}
(hf : ContinuousOn f s) (hg : ContinuousOn g s) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) :
∃ x ∈ s, f x = g x := by
rw [continuousOn_iff_continuous_restrict] at hf hg
obtain ⟨b, h⟩ :=
@intermediate_value_univ₂_eventually₂ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) _ _
(comap_coe_neBot_of_le_principal hl₁) (comap_coe_neBot_of_le_principal hl₂) _ _ hf hg
(he₁.comap _) (he₂.comap _)
exact ⟨b, b.prop, h⟩ | 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.intermediate_value₂_eventually₂ | null |
IsPreconnected.intermediate_value {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a ∈ s)
(hb : b ∈ s) {f : X → α} (hf : ContinuousOn f s) : Icc (f a) (f b) ⊆ f '' s := fun _x hx =>
hs.intermediate_value₂ ha hb hf continuousOn_const hx.1 hx.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 | IsPreconnected.intermediate_value | **Intermediate Value Theorem** for continuous functions on connected sets. |
IsPreconnected.intermediate_value_Ico {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α}
(ht : Tendsto f l (𝓝 v)) : Ico (f a) v ⊆ f '' s := fun _ h =>
hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h.1 (ht.eventually_const_le h.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 | IsPreconnected.intermediate_value_Ico | null |
IsPreconnected.intermediate_value_Ioc {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α}
(ht : Tendsto f l (𝓝 v)) : Ioc v (f a) ⊆ f '' s := fun _ h =>
(hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h.2
(ht.eventually_le_const h.1)).imp fun _ h => h.imp_right Eq.symm | 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.intermediate_value_Ioc | null |
IsPreconnected.intermediate_value_Ioo {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
{v₁ v₂ : α} (ht₁ : Tendsto f l₁ (𝓝 v₁)) (ht₂ : Tendsto f l₂ (𝓝 v₂)) :
Ioo v₁ v₂ ⊆ f '' s := fun _ h =>
hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const
(ht₁.eventually_le_const h.1) (ht₂.eventually_const_le h.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 | IsPreconnected.intermediate_value_Ioo | null |
IsPreconnected.intermediate_value_Ici {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
(ht : Tendsto f l atTop) : Ici (f a) ⊆ f '' s := fun y h =>
hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h (tendsto_atTop.1 ht y) | 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.intermediate_value_Ici | null |
IsPreconnected.intermediate_value_Iic {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
(ht : Tendsto f l atBot) : Iic (f a) ⊆ f '' s := fun y h =>
(hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h (tendsto_atBot.1 ht y)).imp
fun _ h => h.imp_right Eq.symm | 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.intermediate_value_Iic | null |
IsPreconnected.intermediate_value_Ioi {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
{v : α} (ht₁ : Tendsto f l₁ (𝓝 v)) (ht₂ : Tendsto f l₂ atTop) : Ioi v ⊆ f '' s := fun y h =>
hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const
(ht₁.eventually_le_const h) (ht₂.eventually_ge_atTop y) | 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.intermediate_value_Ioi | null |
IsPreconnected.intermediate_value_Iio {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
{v : α} (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ (𝓝 v)) : Iio v ⊆ f '' s := fun y h =>
hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (ht₁.eventually_le_atBot y)
(ht₂.eventually_const_le h) | 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.intermediate_value_Iio | null |
IsPreconnected.intermediate_value_Iii {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
[NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
(ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ atTop) : univ ⊆ f '' s := fun y _ =>
hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (ht₁.eventually_le_atBot y)
(ht₂.eventually_ge_atTop y) | 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.intermediate_value_Iii | null |
intermediate_value_univ [PreconnectedSpace X] (a b : X) {f : X → α} (hf : Continuous f) :
Icc (f a) (f b) ⊆ range f := fun _ hx => intermediate_value_univ₂ hf continuous_const hx.1 hx.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 | intermediate_value_univ | **Intermediate Value Theorem** for continuous functions on connected spaces. |
mem_range_of_exists_le_of_exists_ge [PreconnectedSpace X] {c : α} {f : X → α}
(hf : Continuous f) (h₁ : ∃ a, f a ≤ c) (h₂ : ∃ b, c ≤ f b) : c ∈ range f :=
let ⟨a, ha⟩ := h₁; let ⟨b, hb⟩ := h₂; intermediate_value_univ a b hf ⟨ha, hb⟩
/-! | 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 | mem_range_of_exists_le_of_exists_ge | **Intermediate Value Theorem** for continuous functions on connected spaces. |
IsPreconnected.Icc_subset {s : Set α} (hs : IsPreconnected s) {a b : α} (ha : a ∈ s)
(hb : b ∈ s) : Icc a b ⊆ s := by
simpa only [image_id] using hs.intermediate_value ha hb continuousOn_id | 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.Icc_subset | If a preconnected set contains endpoints of an interval, then it includes the whole interval. |
IsPreconnected.ordConnected {s : Set α} (h : IsPreconnected s) : OrdConnected s :=
⟨fun _ hx _ hy => h.Icc_subset hx hy⟩ | 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.ordConnected | null |
IsConnected.Icc_subset {s : Set α} (hs : IsConnected s) {a b : α} (ha : a ∈ s)
(hb : b ∈ s) : Icc a b ⊆ s :=
hs.2.Icc_subset ha hb | 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_subset | If a preconnected set contains endpoints of an interval, then it includes the whole interval. |
IsPreconnected.eq_univ_of_unbounded {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s)
(ha : ¬BddAbove s) : s = univ := by
refine eq_univ_of_forall fun x => ?_
obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := not_bddBelow_iff.1 hb x
obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bddAbove_iff.1 ha x
exact hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩ | 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.eq_univ_of_unbounded | If preconnected set in a linear order space is unbounded below and above, then it is the whole
space. |
IsConnected.Ioo_csInf_csSup_subset {s : Set α} (hs : IsConnected s) (hb : BddBelow s)
(ha : BddAbove s) : Ioo (sInf s) (sSup s) ⊆ s := fun _x hx =>
let ⟨_y, ys, hy⟩ := (isGLB_lt_iff (isGLB_csInf hs.nonempty hb)).1 hx.1
let ⟨_z, zs, hz⟩ := (lt_isLUB_iff (isLUB_csSup hs.nonempty ha)).1 hx.2
hs.Icc_subset ys zs ⟨hy.le, hz.le⟩ | 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_csInf_csSup_subset | A bounded connected subset of a conditionally complete linear order includes the open interval
`(Inf s, Sup s)`. |
eq_Icc_csInf_csSup_of_connected_bdd_closed {s : Set α} (hc : IsConnected s)
(hb : BddBelow s) (ha : BddAbove s) (hcl : IsClosed s) : s = Icc (sInf s) (sSup s) :=
(subset_Icc_csInf_csSup hb ha).antisymm <|
hc.Icc_subset (hcl.csInf_mem hc.nonempty hb) (hcl.csSup_mem hc.nonempty 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 | eq_Icc_csInf_csSup_of_connected_bdd_closed | null |
IsPreconnected.Ioi_csInf_subset {s : Set α} (hs : IsPreconnected s) (hb : BddBelow s)
(ha : ¬BddAbove s) : Ioi (sInf s) ⊆ s := fun x hx =>
have sne : s.Nonempty := nonempty_of_not_bddAbove ha
let ⟨_y, ys, hy⟩ : ∃ y ∈ s, y < x := (isGLB_lt_iff (isGLB_csInf sne hb)).1 hx
let ⟨_z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bddAbove_iff.1 ha x
hs.Icc_subset ys zs ⟨hy.le, hz.le⟩ | 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.Ioi_csInf_subset | null |
IsPreconnected.Iio_csSup_subset {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s)
(ha : BddAbove s) : Iio (sSup s) ⊆ s :=
IsPreconnected.Ioi_csInf_subset (α := αᵒᵈ) hs ha hb | 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.Iio_csSup_subset | null |
IsPreconnected.mem_intervals {s : Set α} (hs : IsPreconnected s) :
s ∈
({Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s),
Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅} : Set (Set α)) := by
rcases s.eq_empty_or_nonempty with (rfl | hne)
· apply_rules [Or.inr, mem_singleton]
have hs' : IsConnected s := ⟨hne, hs⟩
by_cases hb : BddBelow s <;> by_cases ha : BddAbove s
· refine mem_of_subset_of_mem ?_ <| mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset
(hs'.Ioo_csInf_csSup_subset hb ha) (subset_Icc_csInf_csSup hb ha)
simp only [insert_subset_iff, mem_insert_iff, mem_singleton_iff, true_or, or_true,
singleton_subset_iff, and_self]
· refine Or.inr <| Or.inr <| Or.inr <| Or.inr ?_
rcases mem_Ici_Ioi_of_subset_of_subset (hs.Ioi_csInf_subset hb ha) fun x hx ↦
csInf_le hb hx with hs | hs
· exact Or.inl hs
· exact Or.inr (Or.inl hs)
· iterate 6 apply Or.inr
rcases mem_Iic_Iio_of_subset_of_subset (hs.Iio_csSup_subset hb ha) fun x hx ↦
le_csSup ha hx with hs | hs
· exact Or.inl hs
· exact Or.inr (Or.inl hs)
· iterate 8 apply Or.inr
exact Or.inl (hs.eq_univ_of_unbounded 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 | IsPreconnected.mem_intervals | A preconnected set in a conditionally complete linear order is either one of the intervals
`[Inf s, Sup s]`, `[Inf s, Sup s)`, `(Inf s, Sup s]`, `(Inf s, Sup s)`, `[Inf s, +∞)`,
`(Inf s, +∞)`, `(-∞, Sup s]`, `(-∞, Sup s)`, `(-∞, +∞)`, or `∅`. The converse statement requires
`α` to be densely ordered. |
setOf_isPreconnected_subset_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
intro s hs
rcases hs.mem_intervals with (hs | hs | hs | hs | hs | hs | hs | hs | hs | hs) <;> rw [hs] <;>
simp only [union_insert, union_singleton, mem_insert_iff, mem_union, mem_range, Prod.exists,
uncurry_apply_pair, exists_apply_eq_apply, true_or, or_true, exists_apply_eq_apply2]
/-! | 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_subset_of_ordered | A preconnected set is either one of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`,
`Iic`, `Iio`, or `univ`, or `∅`. The converse statement requires `α` to be densely ordered. Though
one can represent `∅` as `(Inf ∅, Inf ∅)`, we include it into the list of possible cases to improve
readability. |
IsClosed.mem_of_ge_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(ha : a ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty) : b ∈ s := by
let S := s ∩ Icc a b
replace ha : a ∈ S := ⟨ha, left_mem_Icc.2 hab⟩
have Sbd : BddAbove S := ⟨b, fun z hz => hz.2.2⟩
let c := sSup (s ∩ Icc a b)
have c_mem : c ∈ S := hs.csSup_mem ⟨_, ha⟩ Sbd
have c_le : c ≤ b := csSup_le ⟨_, ha⟩ fun x hx => hx.2.2
rcases eq_or_lt_of_le c_le with hc | hc
· exact hc ▸ c_mem.1
exfalso
rcases hgt c ⟨c_mem.1, c_mem.2.1, hc⟩ with ⟨x, xs, cx, xb⟩
exact not_lt_of_ge (le_csSup Sbd ⟨xs, le_trans (le_csSup Sbd ha) (le_of_lt cx), xb⟩) cx | 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 | IsClosed.mem_of_ge_of_forall_exists_gt | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and the set `s ∩ [a, b)` has no maximal point, then `b ∈ s`. |
IsClosed.Icc_subset_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty) : Icc a b ⊆ s := by
intro y hy
have : IsClosed (s ∩ Icc a y) := by
suffices s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y from this ▸ hs.inter isClosed_Icc
grind [inter_assoc, inter_eq_self_of_subset_right, Icc_subset_Icc_right]
exact IsClosed.mem_of_ge_of_forall_exists_gt this ha hy.1 fun x hx ↦
hgt x ⟨hx.1, Ico_subset_Ico_right hy.2 hx.2⟩ y hx.2.2 | 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 | IsClosed.Icc_subset_of_forall_exists_gt | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `a ≤ x < y ≤ b`, `x ∈ s`, the set `s ∩ (x, y]`
is not empty, then `[a, b] ⊆ s`. |
IsClosed.mem_of_ge_of_forall_exists_lt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(hb : b ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty) : a ∈ s := by
suffices OrderDual.toDual a ∈ ofDual ⁻¹' s by aesop
have : IsClosed (OrderDual.ofDual ⁻¹' (s ∩ Icc a b)) := hs
rw [preimage_inter, ←Icc_toDual ] at this
apply this.mem_of_ge_of_forall_exists_gt (by aesop) (by aesop) (fun x hx ↦ ?_)
rw [Ico_toDual, ← preimage_inter, preimage_equiv_eq_image_symm, mem_image] at hx
aesop | 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 | IsClosed.mem_of_ge_of_forall_exists_lt | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `b`, and the set `s ∩ (a, b]` has no minimal point, then `a ∈ s`. |
IsClosed.Icc_subset_of_forall_exists_lt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(hb : b ∈ s) (hgt : ∀ x ∈ s ∩ Ioc a b, ∀ y ∈ Iio x, (s ∩ Ico y x).Nonempty) : Icc a b ⊆ s := by
intro y hy
have : IsClosed (s ∩ Icc y b) := by
suffices s ∩ Icc y b = s ∩ Icc a b ∩ Icc y b from this ▸ hs.inter isClosed_Icc
grind [Icc_subset_Icc_left, inter_eq_self_of_subset_right, inter_assoc]
exact IsClosed.mem_of_ge_of_forall_exists_lt this hb hy.2 fun x hx ↦
hgt x ⟨hx.1, Ioc_subset_Ioc_left hy.1 hx.2⟩ y hx.2.1
variable [DenselyOrdered α] {a b : α} | 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 | IsClosed.Icc_subset_of_forall_exists_lt | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `b`, and for any `a ≤ y < x ≤ b`, `x ∈ s`, the set `s ∩ [y, x)`
is not empty, then `[a, b] ⊆ s`. |
IsClosed.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset {a b : α} {s : Set α}
(hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s)
(h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t) :
Icc a b ⊆ s := by
rcases lt_or_ge b a with hab | hab
· simp_all
set A := {t ∈ Icc a b | Icc a t ⊆ s}
have a_mem : a ∈ A := ⟨left_mem_Icc.mpr hab, by simp [ha]⟩
have bdd_A : BddAbove A := ⟨b, fun t ht ↦ ht.1.2⟩
set t₁ := sSup A
have t₁_mem : t₁ ∈ Icc a b := ⟨le_csSup bdd_A a_mem, csSup_le ⟨a, a_mem⟩ (fun t ht ↦ ht.1.2)⟩
obtain ⟨⟨t₁a, t₁b⟩, ht₁⟩ : t₁ ∈ A := by
refine ⟨t₁_mem, fun t ht ↦ ?_⟩
rcases ht.2.eq_or_lt with rfl | h
· have : closure A ⊆ s ∩ Icc a b := by
apply (closure_subset_iff hs).2 (fun t ht ↦ ⟨?_, ht.1⟩)
have : t ∈ Icc a t := ⟨ht.1.1, le_rfl⟩
exact ht.2 this
apply this.trans inter_subset_left
exact csSup_mem_closure ⟨a, a_mem⟩ bdd_A
· obtain ⟨c, cA, tc⟩ : ∃ c ∈ A, t < c := (lt_csSup_iff bdd_A ⟨a, a_mem⟩).1 h
apply cA.2
exact ⟨ht.1, tc.le⟩
suffices t₁ = b by simpa [this] using ht₁
apply eq_of_le_of_not_lt t₁b fun t₁b' ↦ ?_
obtain ⟨m, t₁m, H⟩ : ∃ m > t₁, Ioo t₁ m ⊆ s :=
(mem_nhdsGT_iff_exists_Ioo_subset' t₁b').mp (h t₁ ⟨t₁a, t₁b'⟩ (fun s hs ↦ ht₁ hs))
obtain ⟨t, hat, ht⟩ : ∃ t, t₁ < t ∧ t < min m b := exists_between (lt_min t₁m t₁b')
have : t ∈ A := by
refine ⟨⟨by order, ht.le.trans (min_le_right _ _)⟩, fun t' ht' ↦ ?_⟩
rcases le_or_gt t' t₁ with h't' | h't'
· exact ht₁ ⟨ht'.1, h't'⟩
· exact H ⟨h't', ht'.2.trans_lt <| ht.trans_le <| min_le_left ..⟩
have : t ≤ t₁ := le_csSup bdd_A this
order | 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 | IsClosed.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `x ∈ [a, b)` such that `[a, x]` is included in `s`,
the set `s` includes some open neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. |
IsClosed.Icc_subset_of_forall_mem_nhdsWithin {a b : α} {s : Set α}
(hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x) :
Icc a b ⊆ s :=
hs.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset ha
(fun _t ht h't ↦ hgt _ ⟨h't ⟨ht.1, le_rfl⟩, ht⟩) | 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 | IsClosed.Icc_subset_of_forall_mem_nhdsWithin | A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `x ∈ s ∩ [a, b)` the set `s` includes some open
neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. |
isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
(ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
(Icc a b ∩ (s ∩ t)).Nonempty := by
have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
by_contra hst
suffices Icc x y ⊆ s from
hst ⟨y, xyab <| right_mem_Icc.2 hxy, this <| right_mem_Icc.2 hxy, hy.2⟩
apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2
rintro z ⟨zs, hz⟩
have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <| Ico_subset_Icc_self hz, zs, zt⟩
have : tᶜ ∩ Ioc z y ∈ 𝓝[>] z := by
rw [← nhdsWithin_Ioc_eq_nhdsGT hz.2]
exact mem_nhdsWithin.2 ⟨tᶜ, ht.isOpen_compl, zt, Subset.rfl⟩
apply mem_of_superset this
have : Ioc z y ⊆ s ∪ t := fun w hw => hab (xyab ⟨le_trans hz.1 (le_of_lt hw.1), hw.2⟩)
exact fun w ⟨wt, wzy⟩ => (this wzy).elim id fun h => (wt h).elim | 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_Icc_aux | null |
isPreconnected_Icc : IsPreconnected (Icc a b) :=
isPreconnected_closed_iff.2
(by
rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩
rcases le_total x y with h | h
· exact isPreconnected_Icc_aux x y s t h hs ht hab hx hy
· rw [inter_comm s t]
rw [union_comm s t] at hab
exact isPreconnected_Icc_aux y x t s h ht hs hab hy hx) | 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_Icc | A closed interval in a densely ordered conditionally complete linear order is preconnected. |
isPreconnected_uIcc : IsPreconnected ([[a, b]]) :=
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 | isPreconnected_uIcc | null |
Set.OrdConnected.isPreconnected {s : Set α} (h : s.OrdConnected) : IsPreconnected s :=
isPreconnected_of_forall_pair fun x hx y hy =>
⟨[[x, y]], h.uIcc_subset hx hy, left_mem_uIcc, right_mem_uIcc, isPreconnected_uIcc⟩ | 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 | Set.OrdConnected.isPreconnected | null |
isPreconnected_iff_ordConnected {s : Set α} : IsPreconnected s ↔ OrdConnected s :=
⟨IsPreconnected.ordConnected, Set.OrdConnected.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_iff_ordConnected | null |
isPreconnected_Ici : IsPreconnected (Ici a) :=
ordConnected_Ici.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_Ici | null |
isPreconnected_Iic : IsPreconnected (Iic a) :=
ordConnected_Iic.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_Iic | null |
isPreconnected_Iio : IsPreconnected (Iio a) :=
ordConnected_Iio.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_Iio | null |
isPreconnected_Ioi : IsPreconnected (Ioi a) :=
ordConnected_Ioi.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_Ioi | null |
isPreconnected_Ioo : IsPreconnected (Ioo a b) :=
ordConnected_Ioo.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_Ioo | null |
isPreconnected_Ioc : IsPreconnected (Ioc a b) :=
ordConnected_Ioc.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_Ioc | null |
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