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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Ico_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ico a d ∈ 𝓝ˢ (Ico b c) | inter_mem (Ici_mem_nhdsSet_Ico h) (Iio_mem_nhdsSet_Ico h') | theorem | Ico_mem_nhdsSet_Ico | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ici_mem_nhdsSet_Ico",
"Iio_mem_nhdsSet_Ico"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioi_mem_nhdsSet_Ioc (h : a ≤ b) : Ioi a ∈ 𝓝ˢ (Ioc b c) | nhdsSet_mono Ioc_subset_Ioi_self <| by simpa | theorem | Ioi_mem_nhdsSet_Ioc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"nhdsSet_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Iio_mem_nhdsSet_Ioc (h : b < c) : Iio c ∈ 𝓝ˢ (Ioc a b) | nhdsSet_mono Ioc_subset_Icc_self <| Iio_mem_nhdsSet_Icc h | theorem | Iio_mem_nhdsSet_Ioc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iio_mem_nhdsSet_Icc",
"nhdsSet_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ici_mem_nhdsSet_Ioc (h : a ≤ b) : Ici a ∈ 𝓝ˢ (Ioc b c) | mem_of_superset (Ioi_mem_nhdsSet_Ioc h) Ioi_subset_Ici_self | theorem | Ici_mem_nhdsSet_Ioc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ioi_mem_nhdsSet_Ioc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Iic_mem_nhdsSet_Ioc (h : b < c) : Iic c ∈ 𝓝ˢ (Ioc a b) | nhdsSet_mono Ioc_subset_Icc_self <| Iic_mem_nhdsSet_Icc h | theorem | Iic_mem_nhdsSet_Ioc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iic_mem_nhdsSet_Icc",
"nhdsSet_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioo_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ioo a d ∈ 𝓝ˢ (Ioc b c) | inter_mem (Ioi_mem_nhdsSet_Ioc h) (Iio_mem_nhdsSet_Ioc h') | theorem | Ioo_mem_nhdsSet_Ioc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iio_mem_nhdsSet_Ioc",
"Ioi_mem_nhdsSet_Ioc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Icc_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Icc a d ∈ 𝓝ˢ (Ioc b c) | inter_mem (Ici_mem_nhdsSet_Ioc h) (Iic_mem_nhdsSet_Ioc h') | theorem | Icc_mem_nhdsSet_Ioc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ici_mem_nhdsSet_Ioc",
"Iic_mem_nhdsSet_Ioc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioc_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ioc a d ∈ 𝓝ˢ (Ioc b c) | inter_mem (Ioi_mem_nhdsSet_Ioc h) (Iic_mem_nhdsSet_Ioc h') | theorem | Ioc_mem_nhdsSet_Ioc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iic_mem_nhdsSet_Ioc",
"Ioi_mem_nhdsSet_Ioc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ico_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ico a d ∈ 𝓝ˢ (Ioc b c) | inter_mem (Ici_mem_nhdsSet_Ioc h) (Iio_mem_nhdsSet_Ioc h') | theorem | Ico_mem_nhdsSet_Ioc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ici_mem_nhdsSet_Ioc",
"Iio_mem_nhdsSet_Ioc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hasBasis_nhdsSet_Iic_Iio (a : α) [h : Nonempty (Ioi a)] :
HasBasis (𝓝ˢ (Iic a)) (a < ·) Iio | by
refine ⟨fun s ↦ ⟨fun hs ↦ ?_, fun ⟨b, hab, hb⟩ ↦ mem_of_superset (Iio_mem_nhdsSet_Iic hab) hb⟩⟩
rw [nhdsSet_Iic, mem_sup, mem_principal] at hs
rcases exists_Ico_subset_of_mem_nhds hs.1 (Set.nonempty_coe_sort.1 h) with ⟨b, hab, hbs⟩
exact ⟨b, hab, Iio_subset_Iio_union_Ico.trans (union_subset hs.2 hbs)⟩ | theorem | hasBasis_nhdsSet_Iic_Iio | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"nhdsSet_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hasBasis_nhdsSet_Iic_Iic (a : α) [NeBot (𝓝[>] a)] :
HasBasis (𝓝ˢ (Iic a)) (a < ·) Iic | by
have : Nonempty (Ioi a) :=
(Filter.nonempty_of_mem (self_mem_nhdsWithin : Ioi a ∈ 𝓝[>] a)).to_subtype
refine (hasBasis_nhdsSet_Iic_Iio _).to_hasBasis
(fun c hc ↦ ?_) (fun _ h ↦ ⟨_, h, Iio_subset_Iic_self⟩)
simpa only [Iic_subset_Iio] using! Filter.nonempty_of_mem (Ioo_mem_nhdsGT hc) | theorem | hasBasis_nhdsSet_Iic_Iic | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Filter.nonempty_of_mem",
"hasBasis_nhdsSet_Iic_Iio",
"self_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Iic_mem_nhdsSet_Iic_iff {a b : α} [NeBot (𝓝[>] b)] : Iic a ∈ 𝓝ˢ (Iic b) ↔ b < a | (hasBasis_nhdsSet_Iic_Iic b).mem_iff.trans
⟨fun ⟨_c, hbc, hca⟩ ↦ hbc.trans_le (Iic_subset_Iic.1 hca), fun h ↦ ⟨_, h, Subset.rfl⟩⟩ | theorem | Iic_mem_nhdsSet_Iic_iff | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"hasBasis_nhdsSet_Iic_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hasBasis_nhdsSet_Ici_Ioi (a : α) [Nonempty (Iio a)] :
HasBasis (𝓝ˢ (Ici a)) (· < a) Ioi | have : Nonempty (Ioi (toDual a)) := ‹_›; hasBasis_nhdsSet_Iic_Iio (toDual a) | theorem | hasBasis_nhdsSet_Ici_Ioi | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"hasBasis_nhdsSet_Iic_Iio"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hasBasis_nhdsSet_Ici_Ici (a : α) [NeBot (𝓝[<] a)] :
HasBasis (𝓝ˢ (Ici a)) (· < a) Ici | have : NeBot (𝓝[>] (toDual a)) := ‹_›; hasBasis_nhdsSet_Iic_Iic (toDual a) | theorem | hasBasis_nhdsSet_Ici_Ici | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"hasBasis_nhdsSet_Iic_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ici_mem_nhdsSet_Ici_iff {a b : α} [NeBot (𝓝[<] b)] : Ici a ∈ 𝓝ˢ (Ici b) ↔ a < b | have : NeBot (𝓝[>] (toDual b)) := ‹_›; Iic_mem_nhdsSet_Iic_iff (a := toDual a) (b := toDual b) | theorem | Ici_mem_nhdsSet_Ici_iff | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iic_mem_nhdsSet_Iic_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
/-- For any `a`, the set `(-∞, a]` is closed. -/
isClosed_Iic (a : α) : IsClosed (Iic a) | class | ClosedIicTopology | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsClosed",
"Preorder",
"TopologicalSpace",
"isClosed_Iic"
] | If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is
closed for all `a : α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ClosedIciTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
/-- For any `a`, the set `[a, +∞)` is closed. -/
isClosed_Ici (a : α) : IsClosed (Ici a) | class | ClosedIciTopology | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsClosed",
"Preorder",
"TopologicalSpace"
] | If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is
closed for all `a : α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
OrderClosedTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
/-- The set `{ (x, y) | x ≤ y }` is a closed set. -/
protected isClosed_le' : IsClosed { p : α × α | p.1 ≤ p.2 } | class | OrderClosedTopology | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsClosed",
"Preorder",
"TopologicalSpace"
] | A topology on a set which is both a topological space and a preorder is _order-closed_ if the
set of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.
This property is satisfied for the order topology on a linear order, but it can be satisfied more
generally, and suffices to der... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) :
Dense (OrderDual.ofDual ⁻¹' s) | hs | theorem | Dense.orderDual | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Dense",
"OrderDual.ofDual",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
BddAbove.of_closure : BddAbove (closure s) → BddAbove s | BddAbove.mono subset_closure | lemma | BddAbove.of_closure | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"BddAbove",
"BddAbove.mono",
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_Iic : IsClosed (Iic a) | ClosedIicTopology.isClosed_Iic a | theorem | isClosed_Iic | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsClosed"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_Iic (a : α) : closure (Iic a) = Iic a | isClosed_Iic.closure_eq | theorem | closure_Iic | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_of_tendsto_of_frequently {x : Filter β} (lim : Tendsto f x (𝓝 a))
(h : ∃ᶠ c in x, f c ≤ b) : a ≤ b | isClosed_Iic.mem_of_frequently_of_tendsto h lim | theorem | le_of_tendsto_of_frequently | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_of_tendsto {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))
(h : ∀ᶠ c in x, f c ≤ b) : a ≤ b | isClosed_Iic.mem_of_tendsto lim h | theorem | le_of_tendsto | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_of_tendsto' {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a))
(h : ∀ c, f c ≤ b) : a ≤ b | le_of_tendsto lim (Eventually.of_forall h) | theorem | le_of_tendsto' | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter",
"le_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
upperBounds_closure (s : Set α) : upperBounds (closure s : Set α) = upperBounds s | ext fun a ↦ by simp_rw [mem_upperBounds_iff_subset_Iic, isClosed_Iic.closure_subset_iff] | lemma | upperBounds_closure | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Set",
"closure",
"mem_upperBounds_iff_subset_Iic",
"upperBounds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
bddAbove_closure : BddAbove (closure s) ↔ BddAbove s | by
simp_rw [BddAbove, upperBounds_closure]
@[to_dual]
protected alias ⟨_, BddAbove.closure⟩ := bddAbove_closure | lemma | bddAbove_closure | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"BddAbove",
"closure",
"upperBounds_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
disjoint_nhds_atBot_iff : Disjoint (𝓝 a) atBot ↔ ¬IsBot a | by
constructor
· intro hd hbot
rw [hbot.atBot_eq, disjoint_principal_right] at hd
exact mem_of_mem_nhds hd le_rfl
· simp only [IsBot, not_forall]
rintro ⟨b, hb⟩
refine disjoint_of_disjoint_of_mem disjoint_compl_left ?_ (Iic_mem_atBot b)
exact isClosed_Iic.isOpen_compl.mem_nhds hb | theorem | disjoint_nhds_atBot_iff | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Disjoint",
"IsBot",
"disjoint_compl_left",
"le_rfl",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.range_of_tendsto {F : Filter β} [F.NeBot] (hle : ∀ i, f i ≤ a)
(hlim : Tendsto f F (𝓝 a)) : IsLUB (range f) a | ⟨forall_mem_range.mpr hle, fun _c hc ↦ le_of_tendsto' hlim fun i ↦ hc <| mem_range_self i⟩ | theorem | IsLUB.range_of_tendsto | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter",
"IsLUB",
"le_of_tendsto'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
disjoint_nhds_atBot (a : α) : Disjoint (𝓝 a) atBot | by simp | theorem | disjoint_nhds_atBot | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Disjoint"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_inf_atBot (a : α) : 𝓝 a ⊓ atBot = ⊥ | (disjoint_nhds_atBot a).eq_bot
@[deprecated (since := "2026-04-07")] alias inf_nhds_atBot := nhds_inf_atBot
@[deprecated (since := "2026-04-07")] alias inf_nhds_atTop := nhds_inf_atTop | theorem | nhds_inf_atBot | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"disjoint_nhds_atBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
not_tendsto_nhds_of_tendsto_atBot (hf : Tendsto f l atBot) (a : α) : ¬Tendsto f l (𝓝 a) | hf.not_tendsto (disjoint_nhds_atBot a).symm | theorem | not_tendsto_nhds_of_tendsto_atBot | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"disjoint_nhds_atBot",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
not_tendsto_atBot_of_tendsto_nhds (hf : Tendsto f l (𝓝 a)) : ¬Tendsto f l atBot | hf.not_tendsto (disjoint_nhds_atBot a) | theorem | not_tendsto_atBot_of_tendsto_nhds | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"disjoint_nhds_atBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iSup_eq_of_forall_le_of_tendsto {ι : Type*} {F : Filter ι} [Filter.NeBot F]
[ConditionallyCompleteLattice α] [TopologicalSpace α] [ClosedIicTopology α]
{a : α} {f : ι → α} (hle : ∀ i, f i ≤ a) (hlim : Filter.Tendsto f F (𝓝 a)) :
⨆ i, f i = a | have := F.nonempty_of_neBot
(IsLUB.range_of_tendsto hle hlim).ciSup_eq | theorem | iSup_eq_of_forall_le_of_tendsto | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ClosedIicTopology",
"ConditionallyCompleteLattice",
"Filter",
"Filter.NeBot",
"Filter.Tendsto",
"IsLUB.range_of_tendsto",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iUnion_Iic_eq_Iio_of_lt_of_tendsto {ι : Type*} {F : Filter ι} [F.NeBot]
[ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIicTopology α]
{a : α} {f : ι → α} (hlt : ∀ i, f i < a) (hlim : Tendsto f F (𝓝 a)) :
⋃ i : ι, Iic (f i) = Iio a | by
have obs : a ∉ range f := by
rw [mem_range]
rintro ⟨i, rfl⟩
exact (hlt i).false
rw [← biUnion_range, (IsLUB.range_of_tendsto (le_of_lt <| hlt ·) hlim).biUnion_Iic_eq_Iio obs] | theorem | iUnion_Iic_eq_Iio_of_lt_of_tendsto | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ClosedIicTopology",
"ConditionallyCompleteLinearOrder",
"Filter",
"IsLUB.range_of_tendsto",
"TopologicalSpace",
"le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_Ioi : IsOpen (Ioi a) | by
rw [← compl_Iic]
exact isClosed_Iic.isOpen_compl | theorem | isOpen_Ioi | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior_Ioi : interior (Ioi a) = Ioi a | isOpen_Ioi.interior_eq | theorem | interior_Ioi | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioi_mem_nhds (h : a < b) : Ioi a ∈ 𝓝 b | IsOpen.mem_nhds isOpen_Ioi h | theorem | Ioi_mem_nhds | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsOpen.mem_nhds",
"isOpen_Ioi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x | Ioi_mem_nhds hab | theorem | eventually_gt_nhds | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioi_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ici_mem_nhds (h : a < b) : Ici a ∈ 𝓝 b | mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self | theorem | Ici_mem_nhds | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioi_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x | Ici_mem_nhds hab | theorem | eventually_ge_nhds | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ici_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.eventually_const_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)
(h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a | h.eventually <| eventually_gt_nhds hv | theorem | Filter.Tendsto.eventually_const_lt | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter",
"Filter.Tendsto",
"eventually_gt_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.eventually_const_le {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)
(h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a | h.eventually <| eventually_ge_nhds hv | theorem | Filter.Tendsto.eventually_const_le | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter",
"eventually_ge_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :
∃ y ∈ s, x < y | hs.exists_mem_open isOpen_Ioi (exists_gt x) | theorem | Dense.exists_gt | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Dense",
"NoMaxOrder",
"Set",
"isOpen_Ioi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :
∃ y ∈ s, x ≤ y | (hs.exists_gt x).imp fun _ h ↦ ⟨h.1, h.2.le⟩ | theorem | Dense.exists_ge | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Dense",
"NoMaxOrder",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :
∃ y ∈ s, x ≤ y | by
by_cases hx : IsTop x
· exact ⟨x, htop x hx, le_rfl⟩
· simp only [IsTop, not_forall, not_le] at hx
rcases hs.exists_mem_open isOpen_Ioi hx with ⟨y, hys, hy : x < y⟩
exact ⟨y, hys, hy.le⟩ | theorem | Dense.exists_ge' | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Dense",
"Set",
"by_cases",
"isOpen_Ioi",
"not_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioo_mem_nhdsLT (H : a < b) : Ioo a b ∈ 𝓝[<] b | by
simpa only [← Iio_inter_Ioi] using inter_mem_nhdsWithin _ (Ioi_mem_nhds H) | theorem | Ioo_mem_nhdsLT | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioi_mem_nhds",
"inter_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioo_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b | mem_of_superset (Ioo_mem_nhdsLT H.1) <| Ioo_subset_Ioo_right H.2 | theorem | Ioo_mem_nhdsLT_of_mem | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioo_mem_nhdsLT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CovBy.nhdsLT (h : a ⋖ b) : 𝓝[<] b = ⊥ | empty_mem_iff_bot.mp <| h.Ioo_eq ▸ Ioo_mem_nhdsLT h.1 | theorem | CovBy.nhdsLT | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioo_mem_nhdsLT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PredOrder.nhdsLT [PredOrder α] : 𝓝[<] a = ⊥ | by
if h : IsMin a then simp [h.Iio_eq]
else exact (Order.pred_covBy_of_not_isMin h).nhdsLT | theorem | PredOrder.nhdsLT | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsMin",
"PredOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PredOrder.nhdsGT_eq_nhdsNE [PredOrder α] (a : α) : 𝓝[>] a = 𝓝[≠] a | by
rw [← nhdsLT_sup_nhdsGT, PredOrder.nhdsLT, bot_sup_eq] | theorem | PredOrder.nhdsGT_eq_nhdsNE | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"PredOrder",
"PredOrder.nhdsLT",
"bot_sup_eq",
"nhdsLT_sup_nhdsGT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PredOrder.nhdsGE_eq_nhds [PredOrder α] (a : α) : 𝓝[≥] a = 𝓝 a | by
rw [← nhdsLT_sup_nhdsGE, PredOrder.nhdsLT, bot_sup_eq] | theorem | PredOrder.nhdsGE_eq_nhds | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"PredOrder",
"PredOrder.nhdsLT",
"bot_sup_eq",
"nhdsLT_sup_nhdsGE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ico_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b | mem_of_superset (Ioo_mem_nhdsLT_of_mem H) Ioo_subset_Ico_self | theorem | Ico_mem_nhdsLT_of_mem | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioo_mem_nhdsLT_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ico_mem_nhdsLT (H : a < b) : Ico a b ∈ 𝓝[<] b | Ico_mem_nhdsLT_of_mem ⟨H, le_rfl⟩ | theorem | Ico_mem_nhdsLT | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ico_mem_nhdsLT_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioc_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b | mem_of_superset (Ioo_mem_nhdsLT_of_mem H) Ioo_subset_Ioc_self | theorem | Ioc_mem_nhdsLT_of_mem | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioo_mem_nhdsLT_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioc_mem_nhdsLT (H : a < b) : Ioc a b ∈ 𝓝[<] b | Ioc_mem_nhdsLT_of_mem ⟨H, le_rfl⟩ | theorem | Ioc_mem_nhdsLT | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioc_mem_nhdsLT_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Icc_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b | mem_of_superset (Ioo_mem_nhdsLT_of_mem H) Ioo_subset_Icc_self | theorem | Icc_mem_nhdsLT_of_mem | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioo_mem_nhdsLT_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Icc_mem_nhdsLT (H : a < b) : Icc a b ∈ 𝓝[<] b | Icc_mem_nhdsLT_of_mem ⟨H, le_rfl⟩ | theorem | Icc_mem_nhdsLT | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Icc_mem_nhdsLT_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_Ico_eq_nhdsLT (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b | nhdsWithin_inter_of_mem <| nhdsWithin_le_nhds <| Ici_mem_nhds h | theorem | nhdsWithin_Ico_eq_nhdsLT | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ici_mem_nhds",
"nhdsWithin_inter_of_mem",
"nhdsWithin_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_Ioo_eq_nhdsLT (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b | nhdsWithin_inter_of_mem <| nhdsWithin_le_nhds <| Ioi_mem_nhds h | theorem | nhdsWithin_Ioo_eq_nhdsLT | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioi_mem_nhds",
"nhdsWithin_inter_of_mem",
"nhdsWithin_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_Ico_iff_Iio (h : a < b) :
ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b | by
simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsLT h] | theorem | continuousWithinAt_Ico_iff_Iio | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ContinuousWithinAt",
"nhdsWithin_Ico_eq_nhdsLT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_Ioo_iff_Iio (h : a < b) :
ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b | by
simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsLT h] | theorem | continuousWithinAt_Ioo_iff_Iio | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ContinuousWithinAt",
"nhdsWithin_Ioo_eq_nhdsLT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CovBy.nhdsLE (H : a ⋖ b) : 𝓝[≤] b = pure b | by
rw [← Iio_insert, nhdsWithin_insert, H.nhdsLT, sup_bot_eq] | theorem | CovBy.nhdsLE | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"nhdsWithin_insert",
"sup_bot_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PredOrder.nhdsLE [PredOrder α] : 𝓝[≤] b = pure b | by
rw [← Iio_insert, nhdsWithin_insert, PredOrder.nhdsLT, sup_bot_eq] | theorem | PredOrder.nhdsLE | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"PredOrder",
"PredOrder.nhdsLT",
"nhdsWithin_insert",
"sup_bot_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioc_mem_nhdsLE (H : a < b) : Ioc a b ∈ 𝓝[≤] b | inter_mem (nhdsWithin_le_nhds <| Ioi_mem_nhds H) self_mem_nhdsWithin | theorem | Ioc_mem_nhdsLE | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioi_mem_nhds",
"nhdsWithin_le_nhds",
"self_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioo_mem_nhdsLE_of_mem (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b | mem_of_superset (Ioc_mem_nhdsLE H.1) <| Ioc_subset_Ioo_right H.2 | theorem | Ioo_mem_nhdsLE_of_mem | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioc_mem_nhdsLE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ico_mem_nhdsLE_of_mem (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b | mem_of_superset (Ioo_mem_nhdsLE_of_mem H) Ioo_subset_Ico_self | theorem | Ico_mem_nhdsLE_of_mem | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioo_mem_nhdsLE_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioc_mem_nhdsLE_of_mem (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b | mem_of_superset (Ioc_mem_nhdsLE H.1) <| Ioc_subset_Ioc_right H.2 | theorem | Ioc_mem_nhdsLE_of_mem | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioc_mem_nhdsLE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Icc_mem_nhdsLE_of_mem (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b | mem_of_superset (Ioc_mem_nhdsLE_of_mem H) Ioc_subset_Icc_self | theorem | Icc_mem_nhdsLE_of_mem | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioc_mem_nhdsLE_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Icc_mem_nhdsLE (H : a < b) : Icc a b ∈ 𝓝[≤] b | Icc_mem_nhdsLE_of_mem ⟨H, le_rfl⟩ | theorem | Icc_mem_nhdsLE | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Icc_mem_nhdsLE_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_Icc_eq_nhdsLE (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b | nhdsWithin_inter_of_mem <| nhdsWithin_le_nhds <| Ici_mem_nhds h | theorem | nhdsWithin_Icc_eq_nhdsLE | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ici_mem_nhds",
"nhdsWithin_inter_of_mem",
"nhdsWithin_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_Ioc_eq_nhdsLE (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b | nhdsWithin_inter_of_mem <| nhdsWithin_le_nhds <| Ioi_mem_nhds h | theorem | nhdsWithin_Ioc_eq_nhdsLE | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Ioi_mem_nhds",
"nhdsWithin_inter_of_mem",
"nhdsWithin_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_Icc_iff_Iic (h : a < b) :
ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b | by
simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsLE h] | theorem | continuousWithinAt_Icc_iff_Iic | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ContinuousWithinAt",
"nhdsWithin_Icc_eq_nhdsLE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_Ioc_iff_Iic (h : a < b) :
ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b | by
simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsLE h] | theorem | continuousWithinAt_Ioc_iff_Iic | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ContinuousWithinAt",
"nhdsWithin_Ioc_eq_nhdsLE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iInf_eq_of_forall_le_of_tendsto {ι : Type*} {F : Filter ι} [F.NeBot]
[ConditionallyCompleteLattice α] [TopologicalSpace α] [ClosedIciTopology α]
{a : α} {f : ι → α} (hle : ∀ i, a ≤ f i) (hlim : Tendsto f F (𝓝 a)) :
⨅ i, f i = a | iSup_eq_of_forall_le_of_tendsto (α := αᵒᵈ) hle hlim | theorem | iInf_eq_of_forall_le_of_tendsto | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ClosedIciTopology",
"ConditionallyCompleteLattice",
"Filter",
"TopologicalSpace",
"iSup_eq_of_forall_le_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iUnion_Ici_eq_Ioi_of_lt_of_tendsto {ι : Type*} {F : Filter ι} [F.NeBot]
[ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIciTopology α]
{a : α} {f : ι → α} (hlt : ∀ i, a < f i) (hlim : Tendsto f F (𝓝 a)) :
⋃ i : ι, Ici (f i) = Ioi a | iUnion_Iic_eq_Iio_of_lt_of_tendsto (α := αᵒᵈ) hlt hlim | theorem | iUnion_Ici_eq_Ioi_of_lt_of_tendsto | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ClosedIciTopology",
"ConditionallyCompleteLinearOrder",
"Filter",
"TopologicalSpace",
"iUnion_Iic_eq_Iio_of_lt_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_le_prod : IsClosed { p : α × α | p.1 ≤ p.2 } | t.isClosed_le' | theorem | isClosed_le_prod | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsClosed"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_le_prod' : IsClosed { p : α × α | p.2 ≤ p.1 } | (isClosed_le_prod (α := α)).preimage continuous_swap | theorem | isClosed_le_prod' | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsClosed",
"continuous_swap",
"isClosed_le_prod"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
IsClosed { b | f b ≤ g b } | continuous_iff_isClosed.mp (hf.prodMk hg) _ isClosed_le_prod | theorem | isClosed_le | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"IsClosed",
"TopologicalSpace",
"isClosed_le_prod"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_Icc {a b : α} : IsClosed (Icc a b) | IsClosed.inter isClosed_Ici isClosed_Iic | theorem | isClosed_Icc | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsClosed",
"IsClosed.inter",
"isClosed_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_Icc (a b : α) : closure (Icc a b) = Icc a b | isClosed_Icc.closure_eq | theorem | closure_Icc | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_of_tendsto_of_tendsto_of_frequently {f g : β → α} {b : Filter β} {a₁ a₂ : α}
(hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∃ᶠ x in b, f x ≤ g x) : a₁ ≤ a₂ | t.isClosed_le'.mem_of_frequently_of_tendsto h (hf.prodMk_nhds hg) | theorem | le_of_tendsto_of_tendsto_of_frequently | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]
(hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ | le_of_tendsto_of_tendsto_of_frequently hf hg <| Eventually.frequently h
@[to_dual self (reorder := f g, a₁ a₂, hf hg)]
alias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto | theorem | le_of_tendsto_of_tendsto | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter",
"le_of_tendsto_of_tendsto_of_frequently"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b]
(hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ | le_of_tendsto_of_tendsto hf hg (Eventually.of_forall h) | theorem | le_of_tendsto_of_tendsto' | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Filter",
"le_of_tendsto_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
closure { b | f b ≤ g b } = { b | f b ≤ g b } | (isClosed_le hf hg).closure_eq | theorem | closure_le_eq | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"TopologicalSpace",
"closure",
"isClosed_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
(hg : Continuous g) : closure { b | f b < g b } ⊆ { b | f b ≤ g b } | (closure_minimal fun _ => le_of_lt) <| isClosed_le hf hg | theorem | closure_lt_subset_le | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"TopologicalSpace",
"closure",
"closure_minimal",
"isClosed_le",
"le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β}
(hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
(h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x | show (f x, g x) ∈ { p : α × α | p.1 ≤ p.2 } from
OrderClosedTopology.isClosed_le'.closure_subset ((hf.prodMk hg).mem_closure hx h) | theorem | ContinuousWithinAt.closure_le | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ContinuousWithinAt",
"Set",
"TopologicalSpace",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s)
(hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s | f x ≤ g x }) | (hf.prodMk hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le' | theorem | IsClosed.isClosed_le | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ContinuousOn",
"IsClosed",
"Set",
"TopologicalSpace"
] | If `s` is a closed set and two functions `f` and `g` are continuous on `s`,
then the set `{x ∈ s | f x ≤ g x}` is a closed set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x)
(hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) :
f x ≤ g x | have : s ⊆ { y ∈ closure s | f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩
(closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2 | theorem | le_on_closure | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ContinuousOn",
"Set",
"TopologicalSpace",
"closure",
"closure_minimal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s)
(hf : ContinuousOn f s) : IsClosed { p : β × α | p.1 ∈ s ∧ f p.1 ≤ p.2 } | (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd | theorem | IsClosed.epigraph | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"ContinuousOn",
"IsClosed",
"Set",
"TopologicalSpace",
"continuousOn_fst",
"continuousOn_snd",
"continuous_fst",
"isClosed_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_monotoneOn [Preorder β] {s : Set β} : IsClosed {f : β → α | MonotoneOn f s} | by
simp only [isClosed_iff_clusterPt, clusterPt_principal_iff_frequently]
intro g hg a ha b hb hab
have hmain (x) : Tendsto (fun f' ↦ f' x) (𝓝 g) (𝓝 (g x)) := continuousAt_apply x _
exact le_of_tendsto_of_tendsto_of_frequently (hmain a) (hmain b) (hg.mono fun g h ↦ h ha hb hab) | theorem | isClosed_monotoneOn | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsClosed",
"MonotoneOn",
"Preorder",
"Set",
"clusterPt_principal_iff_frequently",
"continuousAt_apply",
"isClosed_iff_clusterPt",
"le_of_tendsto_of_tendsto_of_frequently"
] | The set of monotone functions on a set is closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isClosed_monotone [Preorder β] : IsClosed {f : β → α | Monotone f} | by
simp_rw [← monotoneOn_univ]
exact isClosed_monotoneOn | theorem | isClosed_monotone | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsClosed",
"Monotone",
"Preorder",
"isClosed_monotoneOn",
"monotoneOn_univ"
] | The set of monotone functions is closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isClosed_antitoneOn [Preorder β] {s : Set β} : IsClosed {f : β → α | AntitoneOn f s} | isClosed_monotoneOn (α := αᵒᵈ) (β := β) | theorem | isClosed_antitoneOn | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"AntitoneOn",
"IsClosed",
"Preorder",
"Set",
"isClosed_monotoneOn"
] | The set of antitone functions on a set is closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isClosed_antitone [Preorder β] : IsClosed {f : β → α | Antitone f} | isClosed_monotone (α := αᵒᵈ) (β := β) | theorem | isClosed_antitone | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Antitone",
"IsClosed",
"Preorder",
"isClosed_monotone"
] | The set of antitone functions is closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
IsOpen { b | f b < g b } | by
simpa only [lt_iff_not_ge] using! (isClosed_le hg hf).isOpen_compl | theorem | isOpen_lt | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"Continuous",
"IsOpen",
"TopologicalSpace",
"isClosed_le",
"lt_iff_not_ge"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_lt_prod : IsOpen { p : α × α | p.1 < p.2 } | isOpen_lt continuous_fst continuous_snd | theorem | isOpen_lt_prod | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsOpen",
"continuous_fst",
"continuous_snd",
"isOpen_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_Ioo : IsOpen (Ioo a b) | IsOpen.inter isOpen_Ioi isOpen_Iio | theorem | isOpen_Ioo | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsOpen",
"IsOpen.inter",
"isOpen_Ioi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior_Ioo : interior (Ioo a b) = Ioo a b | isOpen_Ioo.interior_eq | theorem | interior_Ioo | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) | by
simp only [interior_Ioo, subset_closure] | theorem | Ioo_subset_closure_interior | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"closure",
"interior",
"interior_Ioo",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x | IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩ | theorem | Ioo_mem_nhds | Topology.Order | Mathlib/Topology/Order/OrderClosed.lean | [] | [
"IsOpen.mem_nhds",
"isOpen_Ioo"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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