statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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image_Icc (hab : a ≤ b) (h : ContinuousOn f <| Icc a b) :
f '' Icc a b = Icc (sInf <| f '' Icc a b) (sSup <| f '' Icc a b) | eq_Icc_of_connected_compact ⟨(nonempty_Icc.2 hab).image f, isPreconnected_Icc.image f h⟩
(isCompact_Icc.image_of_continuousOn h) | theorem | ContinuousOn.image_Icc | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ContinuousOn",
"eq_Icc_of_connected_compact"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_uIcc_eq_Icc (h : ContinuousOn f [[a, b]]) :
f '' [[a, b]] = Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) | image_Icc min_le_max h | theorem | ContinuousOn.image_uIcc_eq_Icc | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ContinuousOn",
"min_le_max"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_uIcc (h : ContinuousOn f <| [[a, b]]) :
f '' [[a, b]] = [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] | by
refine h.image_uIcc_eq_Icc.trans (uIcc_of_le ?_).symm
refine csInf_le_csSup (nonempty_uIcc.image _) ?_ ?_ <;> rw [h.image_uIcc_eq_Icc]
exacts [bddBelow_Icc, bddAbove_Icc] | theorem | ContinuousOn.image_uIcc | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ContinuousOn",
"bddAbove_Icc",
"bddBelow_Icc",
"csInf_le_csSup",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
sInf (f '' Icc a b) ≤ f c | by
have := mem_image_of_mem f hc
rw [h.image_Icc (hc.1.trans hc.2)] at this
exact this.1 | theorem | ContinuousOn.sInf_image_Icc_le | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_sSup_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
f c ≤ sSup (f '' Icc a b) | by
have := mem_image_of_mem f hc
rw [h.image_Icc (hc.1.trans hc.2)] at this
exact this.2 | theorem | ContinuousOn.le_sSup_image_Icc | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_Icc_of_monotoneOn (hab : a ≤ b) (h : ContinuousOn f <| Icc a b)
(h' : MonotoneOn f <| Icc a b) : f '' Icc a b = Icc (f a) (f b) | by
rw [h.image_Icc hab]
congr!
· exact h'.sInf_image_Icc hab
· exact h'.sSup_image_Icc hab | theorem | ContinuousOn.image_Icc_of_monotoneOn | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"ContinuousOn",
"MonotoneOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_Icc_of_antitoneOn (hab : a ≤ b) (h : ContinuousOn f <| Icc a b)
(h' : AntitoneOn f <| Icc a b) : f '' Icc a b = Icc (f b) (f a) | by
have : Icc (f b) (f a) = Icc (toDual (f a)) (toDual (f b)) := by rw [Icc_toDual]; rfl
rw [this]
exact image_Icc_of_monotoneOn (β := βᵒᵈ) hab h h'.dual_right | theorem | ContinuousOn.image_Icc_of_antitoneOn | Topology.Order | Mathlib/Topology/Order/Compact.lean | [] | [
"AntitoneOn",
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
range_Iio : HasCountableSeparatingOn X (· ∈ range Iio) s | by
constructor
rcases TopologicalSpace.exists_countable_dense X with ⟨s, hsc, hsd⟩
set t := s ∪ {x | ∃ y, y ⋖ x}
refine ⟨Iio '' t, .image ?_ _, ?_, ?_⟩
· exact hsc.union countable_setOf_covBy_left
· exact image_subset_range _ _
· rintro x - y - h
by_contra! hne
wlog hlt : x < y generalizing x y
... | instance | HasCountableSeparatingOn.range_Iio | Topology.Order | Mathlib/Topology/Order/CountableSeparating.lean | [] | [
"HasCountableSeparatingOn",
"TopologicalSpace.exists_countable_dense",
"isOpen_Ioo"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
range_Ioi : HasCountableSeparatingOn X (· ∈ range Ioi) s | .range_Iio (X := Xᵒᵈ) | instance | HasCountableSeparatingOn.range_Ioi | Topology.Order | Mathlib/Topology/Order/CountableSeparating.lean | [] | [
"HasCountableSeparatingOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
range_Iic : HasCountableSeparatingOn X (· ∈ range Iic) s | let ⟨t, htc, ht_sub, ht⟩ := (range_Ioi (X := X) (s := s)).1
⟨compl '' t, htc.image _, by simpa [← compl_inj_iff (x := Ioi _)] using ht_sub,
by simpa [not_iff_not]⟩ | instance | HasCountableSeparatingOn.range_Iic | Topology.Order | Mathlib/Topology/Order/CountableSeparating.lean | [] | [
"HasCountableSeparatingOn",
"compl_inj_iff",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
range_Ici : HasCountableSeparatingOn X (· ∈ range Ici) s | range_Iic (X := Xᵒᵈ) | instance | HasCountableSeparatingOn.range_Ici | Topology.Order | Mathlib/Topology/Order/CountableSeparating.lean | [] | [
"HasCountableSeparatingOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_forall_eventually_lt_iff (h : ∀ x, ∀ᶠ a in l, f a < x ↔ g a < x) : f =ᶠ[l] g | of_forall_separating_preimage (· ∈ range Iio) <| forall_mem_range.2 <| fun x ↦ .set_eq (h x) | lemma | Filter.EventuallyEq.of_forall_eventually_lt_iff | Topology.Order | Mathlib/Topology/Order/CountableSeparating.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_forall_eventually_le_iff (h : ∀ x, ∀ᶠ a in l, f a ≤ x ↔ g a ≤ x) : f =ᶠ[l] g | of_forall_separating_preimage (· ∈ range Iic) <| forall_mem_range.2 <| fun x ↦ .set_eq (h x) | lemma | Filter.EventuallyEq.of_forall_eventually_le_iff | Topology.Order | Mathlib/Topology/Order/CountableSeparating.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_forall_eventually_gt_iff (h : ∀ x, ∀ᶠ a in l, x < f a ↔ x < g a) : f =ᶠ[l] g | of_forall_eventually_lt_iff (X := Xᵒᵈ) h | lemma | Filter.EventuallyEq.of_forall_eventually_gt_iff | Topology.Order | Mathlib/Topology/Order/CountableSeparating.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_forall_eventually_ge_iff (h : ∀ x, ∀ᶠ a in l, x ≤ f a ↔ x ≤ g a) : f =ᶠ[l] g | of_forall_eventually_le_iff (X := Xᵒᵈ) h | lemma | Filter.EventuallyEq.of_forall_eventually_ge_iff | Topology.Order | Mathlib/Topology/Order/CountableSeparating.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a | by
apply Subset.antisymm
· exact closure_minimal Ioi_subset_Ici_self isClosed_Ici
· rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]
exact isGLB_Ioi.mem_closure h | theorem | closure_Ioi' | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure",
"closure_minimal",
"diff_subset_closure_iff"
] | The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a | closure_Ioi' nonempty_Ioi | theorem | closure_Ioi | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"closure",
"closure_Ioi'"
] | The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a | closure_Ioi' (α := αᵒᵈ) h | theorem | closure_Iio' | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure",
"closure_Ioi'"
] | The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom
element. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a | closure_Iio' nonempty_Iio | theorem | closure_Iio | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMinOrder",
"closure",
"closure_Iio'"
] | The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsMax.of_disjoint_nhds_Ioi {x : α} {u : Set α} (hu : u ∈ nhds x)
(hd : Disjoint u (Set.Ioi x)) : IsMax x | by
by_contra hx
exact (mem_closure_iff_nhds.mp (closure_Ioi' (not_isMax_iff.mp hx) ▸ self_mem_Ici) u hu).ne_empty
(disjoint_iff.mp hd) | theorem | IsMax.of_disjoint_nhds_Ioi | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"Disjoint",
"Set",
"closure_Ioi'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsMin.of_disjoint_nhds_Iio {x : α} {u : Set α} (hu : u ∈ nhds x)
(hd : Disjoint u (Set.Iio x)) : IsMin x | IsMax.of_disjoint_nhds_Ioi (α := αᵒᵈ) hu hd | theorem | IsMin.of_disjoint_nhds_Iio | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"Disjoint",
"IsMax.of_disjoint_nhds_Ioi",
"IsMin",
"Set",
"Set.Iio"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty_nhds_inter_Ioi {x : α} {u : Set α} (hu : u ∈ nhds x) (hx : ¬IsMax x) :
(u ∩ Set.Ioi x).Nonempty | by
by_contra h
exact hx (IsMax.of_disjoint_nhds_Ioi hu (Set.disjoint_iff_inter_eq_empty.mpr
(Set.not_nonempty_iff_eq_empty.mp h))) | theorem | nonempty_nhds_inter_Ioi | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"IsMax.of_disjoint_nhds_Ioi",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty_nhds_inter_Iio {x : α} {u : Set α} (hu : u ∈ nhds x) (hx : ¬IsMin x) :
(u ∩ Set.Iio x).Nonempty | nonempty_nhds_inter_Ioi (α := αᵒᵈ) hu hx | theorem | nonempty_nhds_inter_Iio | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"IsMin",
"Set",
"Set.Iio",
"nonempty_nhds_inter_Ioi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b | by
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· rcases hab.lt_or_gt with hab | hab
· rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]
have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab
simp only [insert_subset_iff, singleton_subset_iff]
exact ⟨(i... | theorem | closure_Ioo | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure",
"closure_minimal",
"diff_subset_closure_iff",
"isClosed_Icc",
"isGLB_Ioo",
"isLUB_Ioo"
] | The closure of the open interval `(a, b)` is the closed interval `[a, b]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
closure_uIoo {a b : α} (hab : a ≠ b) : closure (uIoo a b) = uIcc a b | by
simp [uIoo, uIcc, hab] | theorem | closure_uIoo | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b | by
apply Subset.antisymm
· exact closure_minimal Ioc_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)
rw [closure_Ioo hab] | theorem | closure_Ioc | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure",
"closure_Ioo",
"closure_minimal",
"closure_mono",
"isClosed_Icc"
] | The closure of the interval `(a, b]` is the closed interval `[a, b]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
closure_uIoc {a b : α} (hab : a ≠ b) : closure (uIoc a b) = uIcc a b | by
simp [uIoc, uIcc, hab] | theorem | closure_uIoc | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b | by
apply Subset.antisymm
· exact closure_minimal Ico_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)
rw [closure_Ioo hab] | theorem | closure_Ico | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure",
"closure_Ioo",
"closure_minimal",
"closure_mono",
"isClosed_Icc"
] | The closure of the interval `[a, b)` is the closed interval `[a, b]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a | by
rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic] | theorem | interior_Ici' | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure_Iio'",
"interior",
"interior_compl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a | interior_Ici' nonempty_Iio | theorem | interior_Ici | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMinOrder",
"interior",
"interior_Ici'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a | interior_Ici' (α := αᵒᵈ) ha | theorem | interior_Iic' | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"interior",
"interior_Ici'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a | interior_Iic' nonempty_Ioi | theorem | interior_Iic | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"interior",
"interior_Iic'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b | by
rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio] | theorem | interior_Icc | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"NoMinOrder",
"interior",
"interior_Ici",
"interior_Iic",
"interior_inter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Icc_mem_nhds_iff [NoMinOrder α] [NoMaxOrder α] {a b x : α} :
Icc a b ∈ 𝓝 x ↔ x ∈ Ioo a b | by
rw [← interior_Icc, mem_interior_iff_mem_nhds] | theorem | Icc_mem_nhds_iff | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"NoMinOrder",
"interior_Icc",
"mem_interior_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b | by
rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio] | theorem | interior_Ico | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMinOrder",
"interior",
"interior_Ici",
"interior_inter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ico_mem_nhds_iff [NoMinOrder α] {a b x : α} : Ico a b ∈ 𝓝 x ↔ x ∈ Ioo a b | by
rw [← interior_Ico, mem_interior_iff_mem_nhds] | theorem | Ico_mem_nhds_iff | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMinOrder",
"interior_Ico",
"mem_interior_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b | by
rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio] | theorem | interior_Ioc | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"interior",
"interior_Iic",
"interior_Ioi",
"interior_inter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioc_mem_nhds_iff [NoMaxOrder α] {a b x : α} : Ioc a b ∈ 𝓝 x ↔ x ∈ Ioo a b | by
rw [← interior_Ioc, mem_interior_iff_mem_nhds] | theorem | Ioc_mem_nhds_iff | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"interior_Ioc",
"mem_interior_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b | (closure_minimal interior_subset isClosed_Icc).antisymm <|
calc
Icc a b = closure (Ioo a b) := (closure_Ioo h).symm
_ ⊆ closure (interior (Icc a b)) :=
closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo) | theorem | closure_interior_Icc | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"antisymm",
"closure",
"closure_Ioo",
"closure_minimal",
"closure_mono",
"interior",
"interior_maximal",
"interior_subset",
"isClosed_Icc",
"isOpen_Ioo",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) | by
rcases eq_or_ne a b with (rfl | h)
· simp
· calc
Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self
_ = closure (Ioo a b) := (closure_Ioo h).symm
_ ⊆ closure (interior (Ioc a b)) :=
closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo) | theorem | Ioc_subset_closure_interior | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure",
"closure_Ioo",
"closure_mono",
"eq_or_ne",
"interior",
"interior_maximal",
"isOpen_Ioo",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) | by
simpa only [Ioc_toDual] using!
Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a) | theorem | Ico_subset_closure_interior | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"Ioc_subset_closure_interior",
"OrderDual.toDual",
"closure",
"interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} | by
simp [frontier, ha] | theorem | frontier_Ici' | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"frontier"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} | frontier_Ici' nonempty_Iio | theorem | frontier_Ici | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMinOrder",
"frontier",
"frontier_Ici'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} | by
simp [frontier, ha] | theorem | frontier_Iic' | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"frontier"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} | frontier_Iic' nonempty_Ioi | theorem | frontier_Iic | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"frontier",
"frontier_Iic'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} | by
simp [frontier, closure_Ioi' ha] | theorem | frontier_Ioi' | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure_Ioi'",
"frontier"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} | frontier_Ioi' nonempty_Ioi | theorem | frontier_Ioi | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"frontier",
"frontier_Ioi'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} | by
simp [frontier, closure_Iio' ha] | theorem | frontier_Iio' | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure_Iio'",
"frontier"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} | frontier_Iio' nonempty_Iio | theorem | frontier_Iio | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMinOrder",
"frontier",
"frontier_Iio'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :
frontier (Icc a b) = {a, b} | by simp [frontier, h, Icc_diff_Ioo_same] | theorem | frontier_Icc | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"NoMinOrder",
"frontier"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} | by
rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le] | theorem | frontier_Ioo | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure_Ioo",
"frontier",
"interior_Ioo"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} | by
rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le] | theorem | frontier_Ico | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMinOrder",
"closure_Ico",
"frontier",
"interior_Ico"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} | by
rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le] | theorem | frontier_Ioc | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"closure_Ioc",
"frontier",
"interior_Ioc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :
NeBot (𝓝[Ioi a] b) | mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁] | theorem | nhdsWithin_Ioi_neBot' | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure_Ioi'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) | nhdsWithin_Ioi_neBot' nonempty_Ioi H | theorem | nhdsWithin_Ioi_neBot | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"nhdsWithin_Ioi_neBot'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsGT_neBot_of_exists_gt {a : α} (H : ∃ b, a < b) : NeBot (𝓝[>] a) | nhdsWithin_Ioi_neBot' H (le_refl a) | theorem | nhdsGT_neBot_of_exists_gt | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"le_refl",
"nhdsWithin_Ioi_neBot'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsGT_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) | nhdsWithin_Ioi_neBot le_rfl | instance | nhdsGT_neBot | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMaxOrder",
"le_rfl",
"nhdsWithin_Ioi_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :
NeBot (𝓝[Iio c] b) | mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁] | theorem | nhdsWithin_Iio_neBot' | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"closure_Iio'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) | nhdsWithin_Iio_neBot' nonempty_Iio H | theorem | nhdsWithin_Iio_neBot | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMinOrder",
"nhdsWithin_Iio_neBot'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsLT_neBot_of_exists_lt {b : α} (H : ∃ a, a < b) : NeBot (𝓝[<] b) | nhdsWithin_Iio_neBot' H (le_refl b)
@[deprecated (since := "2026-01-16")] alias nhdsWithin_Iio_self_neBot' := nhdsLT_neBot_of_exists_lt | theorem | nhdsLT_neBot_of_exists_lt | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"le_refl",
"nhdsWithin_Iio_neBot'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsLT_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) | nhdsWithin_Iio_neBot (le_refl a) | instance | nhdsLT_neBot | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"NoMinOrder",
"le_refl",
"nhdsWithin_Iio_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) | (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H) | theorem | right_nhdsWithin_Ico_neBot | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"isLUB_Ico",
"nhdsWithin_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) | (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H) | theorem | left_nhdsWithin_Ioc_neBot | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"isGLB_Ioc",
"nhdsWithin_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) | (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H) | theorem | left_nhdsWithin_Ioo_neBot | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"isGLB_Ioo",
"nhdsWithin_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) | (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H) | theorem | right_nhdsWithin_Ioo_neBot | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"isLUB_Ioo",
"nhdsWithin_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
DenselyOrdered.subsingleton_of_discreteTopology [DiscreteTopology α] : Subsingleton α | by
suffices ∀ a b : α, b ≤ a from ⟨fun a b ↦ le_antisymm (this b a) (this a b)⟩
intro a b
by_contra! contra
have : Ioo a b = Icc a b := by rw [← closure_discrete (Ioo a b), closure_Ioo contra.ne]
grind => have : b ∈ Ioo a b; finish | lemma | DenselyOrdered.subsingleton_of_discreteTopology | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"DiscreteTopology",
"closure_Ioo",
"closure_discrete",
"le_antisymm"
] | If the order topology for a dense linear ordering is discrete, the space has at most one point.
We would prefer for this to be an instance but even at `(priority := 100)` this was problematic so
we have deferred this issue. TODO Promote this to an `instance`! | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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)... | theorem | Dense.exists_countable_dense_subset_no_bot_top | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"Dense",
"IsBot",
"Nontrivial",
"Set"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | exists_countable_dense_no_bot_top | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"Dense",
"IsBot",
"Nontrivial",
"Set"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | isClosed_Ico_iff | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"IsClosed",
"Set.Ico",
"closure_Ico",
"le_of_not_gt"
] | `Set.Ico a b` is only closed if it is empty. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | isClosed_Ioc_iff | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"IsClosed",
"Set.Ioc",
"closure_Ioc",
"le_of_not_gt"
] | `Set.Ioc a b` is only closed if it is empty. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | isClosed_Ioo_iff | Topology.Order | Mathlib/Topology/Order/DenselyOrdered.lean | [] | [
"IsClosed",
"Set.Ioo",
"closure_Ioo",
"le_of_not_gt"
] | `Set.Ioo a b` is only closed if it is empty. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousOn_Icc_extendFrom_Ioo
(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
by_cases! hab : a = b
· simp [hab]
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_subse... | theorem | continuousOn_Icc_extendFrom_Ioo | Topology.Order | Mathlib/Topology/Order/ExtendFrom.lean | [] | [
"ContinuousOn",
"closure_Ioo",
"continuousOn_extendFrom",
"extendFrom",
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousOn_uIcc_extendFrom_uIoo
(hf : ContinuousOn f (uIoo a b))
(ha : Tendsto f (𝓝[uIoo a b] a) (𝓝 la)) (hb : Tendsto f (𝓝[uIoo a b] b) (𝓝 lb)) :
ContinuousOn (extendFrom (uIoo a b) f) (uIcc a b) | by
by_cases! hab : a = b
· simp [hab]
obtain hab' | hba' := hab.lt_or_gt
· simp only [hab', uIoo_of_lt, nhdsWithin_Ioo_eq_nhdsGT, nhdsWithin_Ioo_eq_nhdsLT,
uIcc_of_lt] at ha hb hf ⊢
exact continuousOn_Icc_extendFrom_Ioo hf ha hb
· simp only [hba', uIoo_of_gt, nhdsWithin_Ioo_eq_nhdsGT, nhdsWithin_Ioo... | theorem | continuousOn_uIcc_extendFrom_uIoo | Topology.Order | Mathlib/Topology/Order/ExtendFrom.lean | [] | [
"ContinuousOn",
"continuousOn_Icc_extendFrom_Ioo",
"extendFrom",
"nhdsWithin_Ioo_eq_nhdsLT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousOn_Ico_extendFrom_Ioo
(hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la)) :
ContinuousOn (extendFrom (Ioo a b) f) (Ico a b) | by
by_cases! hab : a ≥ b
· simp [hab]
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 | continuousOn_Ico_extendFrom_Ioo | Topology.Order | Mathlib/Topology/Order/ExtendFrom.lean | [] | [
"ContinuousOn",
"closure_Ioo",
"continuousOn_extendFrom",
"extendFrom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousOn_Ioc_extendFrom_Ioo
(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 (f := f ∘ OrderDual.ofDual) (a := OrderDual.toDual b)
(b := OrderDual.toDual a) (la := lb)
rw [Ico_toDual, Ioi_toDual, Ioo_toDual] at this
exact this hf hb | theorem | continuousOn_Ioc_extendFrom_Ioo | Topology.Order | Mathlib/Topology/Order/ExtendFrom.lean | [] | [
"ContinuousOn",
"OrderDual.ofDual",
"OrderDual.toDual",
"continuousOn_Ico_extendFrom_Ioo",
"extendFrom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_lim_at_left_extendFrom_Ioo (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 | eq_lim_at_left_extendFrom_Ioo | Topology.Order | Mathlib/Topology/Order/ExtendFrom.lean | [] | [
"closure_Ioo",
"extendFrom",
"extendFrom_eq",
"le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_lim_at_right_extendFrom_Ioo (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 | eq_lim_at_right_extendFrom_Ioo | Topology.Order | Mathlib/Topology/Order/ExtendFrom.lean | [] | [
"closure_Ioo",
"extendFrom",
"extendFrom_eq",
"le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_lim_at_left_extendFrom_uIoo (hab : a ≠ b)
(ha : Tendsto f (𝓝[uIoo a b] a) (𝓝 la)) : extendFrom (uIoo a b) f a = la | extendFrom_eq (by simp [hab]) ha | theorem | eq_lim_at_left_extendFrom_uIoo | Topology.Order | Mathlib/Topology/Order/ExtendFrom.lean | [] | [
"extendFrom",
"extendFrom_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_lim_at_right_extendFrom_uIoo (hab : a ≠ b)
(hb : Tendsto f (𝓝[uIoo a b] b) (𝓝 lb)) : extendFrom (uIoo a b) f b = lb | extendFrom_eq (by simp [hab]) hb | theorem | eq_lim_at_right_extendFrom_uIoo | Topology.Order | Mathlib/Topology/Order/ExtendFrom.lean | [] | [
"extendFrom",
"extendFrom_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsMaxOn.closure | Topology.Order | Mathlib/Topology/Order/ExtrClosure.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.closure_le",
"IsMaxOn",
"closure",
"continuousWithinAt_const",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsMinOn.closure (h : IsMinOn f s a) (hc : ContinuousOn f (closure s)) :
IsMinOn f (closure s) a | h.dual.closure hc | theorem | IsMinOn.closure | Topology.Order | Mathlib/Topology/Order/ExtrClosure.lean | [] | [
"ContinuousOn",
"IsMinOn",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsExtrOn.closure | Topology.Order | Mathlib/Topology/Order/ExtrClosure.lean | [] | [
"ContinuousOn",
"IsExtrOn",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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_... | theorem | IsLocalMaxOn.closure | Topology.Order | Mathlib/Topology/Order/ExtrClosure.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt.closure_le",
"IsLocalMaxOn",
"closure",
"continuousWithinAt_const",
"mem_closure_iff_nhdsWithin_neBot",
"nhdsWithin_inter_of_mem",
"nhdsWithin_le_nhds",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLocalMinOn.closure (h : IsLocalMinOn f s a) (hc : ContinuousOn f (closure s)) :
IsLocalMinOn f (closure s) a | IsLocalMaxOn.closure h.dual hc | theorem | IsLocalMinOn.closure | Topology.Order | Mathlib/Topology/Order/ExtrClosure.lean | [] | [
"ContinuousOn",
"IsLocalMaxOn.closure",
"IsLocalMinOn",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsLocalExtrOn.closure | Topology.Order | Mathlib/Topology/Order/ExtrClosure.lean | [] | [
"ContinuousOn",
"IsLocalExtrOn",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Filter.tendsto_nhds_atTop | Topology.Order | Mathlib/Topology/Order/Filter.lean | [] | [
"Filter",
"NoMaxOrder",
"le_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_atBot [NoMinOrder X] : Tendsto 𝓝 (atBot : Filter X) (𝓝 atBot) | @Filter.tendsto_nhds_atTop Xᵒᵈ _ _ _ _ | theorem | Filter.tendsto_nhds_atBot | Topology.Order | Mathlib/Topology/Order/Filter.lean | [] | [
"Filter",
"Filter.tendsto_nhds_atTop",
"NoMinOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Filter.Tendsto.nhds_atTop | Topology.Order | Mathlib/Topology/Order/Filter.lean | [] | [
"Filter",
"NoMaxOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Filter.Tendsto.nhds_atBot | Topology.Order | Mathlib/Topology/Order/Filter.lean | [] | [
"Filter",
"NoMinOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hull (T : Set α) (a : α) | T ↓∩ Ici a | abbrev | PrimitiveSpectrum.hull | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"Set"
] | For `a` of type `α` the set of element of `T` which dominate `a` is the `hull` of `a` in `T`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
hull_inf (hT : ∀ p ∈ T, InfPrime p) (a b : α) :
hull T (a ⊓ b) = hull T a ∪ hull T b | by
grind [InfPrime.inf_le] | lemma | PrimitiveSpectrum.hull_inf | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"InfPrime",
"InfPrime.inf_le"
] | The set of relative-closed sets of the form `hull T a` for some `a` in `α` is closed under
pairwise union. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
hull_finsetInf (hT : ∀ p ∈ T, InfPrime p) (F : Finset α) :
hull T (inf F id) = T ↓∩ upperClosure (F : Set α) | 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 (Sub... | lemma | PrimitiveSpectrum.hull_finsetInf | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"Finset",
"Finset.cons_induction",
"InfPrime",
"Set",
"Set.not_nonempty_iff_eq_empty",
"Set.preimage_empty",
"Subtype.coe_prop",
"coe_upperClosure",
"upperClosure"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
preimage_upperClosure_compl_finset (hT : ∀ p ∈ T, InfPrime p) (F : Finset α) :
T ↓∩ (upperClosure (F : Set α))ᶜ = (hull T (inf F id))ᶜ | by
rw [Set.preimage_compl, (hull_finsetInf hT)] | lemma | PrimitiveSpectrum.preimage_upperClosure_compl_finset | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"Finset",
"InfPrime",
"Set",
"Set.preimage_compl",
"upperClosure"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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_injec... | lemma | PrimitiveSpectrum.isTopologicalBasis_relativeLower | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"Finset",
"Function.Injective.preimage_image",
"InfPrime",
"Set",
"Subtype.coe_preimage_self",
"Subtype.image_preimage_coe",
"Subtype.val_injective",
"Topology.IsLower.isTopologicalBasis",
"UpperSet.coe_Ici",
"exists_exists_and_eq_and",
"isTopologicalBasis_subtype",
"upperClosure_singleton"
] | The relative-open sets of the form `(hull T a)ᶜ` for `a` in `α` form a basis for the relative
Lower topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
hull_iSup {ι : Sort v} (s : ι → α) : hull T (iSup s) = ⋂ i, hull T (s i) | by aesop | lemma | PrimitiveSpectrum.hull_iSup | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"iSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hull_sSup (S : Set α) : hull T (sSup S) = ⋂₀ { hull T a | a ∈ S } | by aesop | lemma | PrimitiveSpectrum.hull_sSup | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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⟩ | lemma | PrimitiveSpectrum.isOpen_iff | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"InfPrime",
"IsOpen",
"Set",
"TopologicalSpace"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | PrimitiveSpectrum.isClosed_iff | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"InfPrime",
"IsClosed",
"Set",
"TopologicalSpace",
"compl_inj_iff",
"isOpen_compl_iff"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
kernel (S : Set T) | sInf (Subtype.val '' S) | abbrev | PrimitiveSpectrum.kernel | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"Set"
] | For a subset `S` of `T`, `kernel S` is the infimum of `S` (considered as a set of `α`) | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
gc : GaloisConnection (α := Set T) (β := αᵒᵈ)
(fun S => toDual (kernel S)) (fun a => hull T (ofDual a)) | fun S a => by
simp [Set.subset_def] | theorem | PrimitiveSpectrum.gc | Topology.Order | Mathlib/Topology/Order/HullKernel.lean | [] | [
"GaloisConnection",
"Set",
"Set.subset_def"
] | The pair of maps `kernel` and `hull` form an antitone Galois connection between the
subsets of `T` and `α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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