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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tendsto_rightLim_of_tendsto [TopologicalSpace α] [OrderTopology α]
{f : α → β} {a : α} (h : ∃ y, Tendsto f (𝓝[>] a) (𝓝 y)) :
Tendsto f (𝓝[>] a) (𝓝 (f.rightLim a)) | tendsto_leftLim_of_tendsto (α := αᵒᵈ) h | theorem | tendsto_rightLim_of_tendsto | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"TopologicalSpace",
"tendsto_leftLim_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mapClusterPt_leftLim [TopologicalSpace α] [OrderTopology α]
(f : α → β) (a : α) : MapClusterPt (f.leftLim a) (𝓝[≤] a) f | by
have A : (𝓝 (f a) ⊓ map f (𝓝[≤] a)).NeBot := by
refine inf_neBot_iff.mpr (fun s hs s' hs' ↦ ?_)
refine ⟨f a, mem_of_mem_nhds hs, ?_⟩
simp only [mem_map] at hs'
apply mem_of_mem_nhdsWithin self_mem_Iic hs'
rcases eq_or_neBot (𝓝[<] a) with h' | h'
· simp only [MapClusterPt, ClusterPt, h', left... | theorem | mapClusterPt_leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ClusterPt",
"MapClusterPt",
"MapClusterPt.mono",
"OrderTopology",
"TopologicalSpace",
"leftLim_eq_of_eq_bot",
"leftLim_eq_of_not_tendsto",
"mem_of_mem_nhds",
"mem_of_mem_nhdsWithin",
"nhdsWithin_mono",
"tendsto_leftLim_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mapClusterPt_rightLim [TopologicalSpace α] [OrderTopology α]
(f : α → β) (a : α) : MapClusterPt (f.rightLim a) (𝓝[≥] a) f | mapClusterPt_leftLim (α := αᵒᵈ) _ _ | theorem | mapClusterPt_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"MapClusterPt",
"OrderTopology",
"TopologicalSpace",
"mapClusterPt_leftLim"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_leftLim_Iic [TopologicalSpace α] [OrderTopology α] [T3Space β]
{f : α → β} {a : α} (h : Tendsto f (𝓝[<] a) (𝓝 (f.leftLim a))) :
ContinuousWithinAt f.leftLim (Iic a) a | by
have : 𝓝[≤] a = 𝓝[<] a ⊔ pure a := by
rw [← Iio_union_Icc_eq_Iic le_rfl, nhdsWithin_union]
simp
rw [ContinuousWithinAt, this, tendsto_sup]
simp only [tendsto_pure_nhds, and_true]
apply (closed_nhds_basis (f.leftLim a)).tendsto_right_iff.2
rintro s ⟨s_mem, s_closed⟩
rcases eq_or_neBot (𝓝[<] a) ... | theorem | continuousWithinAt_leftLim_Iic | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ContinuousWithinAt",
"Filter.nonempty_of_mem",
"Ioo_mem_nhdsLT",
"Ioo_mem_nhdsLT_of_mem",
"OrderTopology",
"T3Space",
"TopologicalSpace",
"closed_nhds_basis",
"le_rfl",
"leftLim_eq_of_eq_bot",
"leftLim_eq_of_not_tendsto",
"mem_nhdsLT_iff_exists_Ioo_subset'",
"nhdsWithin_union",
"self_mem_... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim_leftLim [TopologicalSpace α] [OrderTopology α] [T3Space β]
{f : α → β} {a : α} (h : Tendsto f (𝓝[<] a) (𝓝 (f.leftLim a))) :
f.leftLim.leftLim a = f.leftLim a | (continuousWithinAt_leftLim_Iic h).leftLim_eq | theorem | leftLim_leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"T3Space",
"TopologicalSpace",
"continuousWithinAt_leftLim_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_rightLim_Ici [TopologicalSpace α] [OrderTopology α] [T3Space β]
{f : α → β} {a : α} (h : Tendsto f (𝓝[>] a) (𝓝 (f.rightLim a))) :
ContinuousWithinAt f.rightLim (Ici a) a | continuousWithinAt_leftLim_Iic (α := αᵒᵈ) h | theorem | continuousWithinAt_rightLim_Ici | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ContinuousWithinAt",
"OrderTopology",
"T3Space",
"TopologicalSpace",
"continuousWithinAt_leftLim_Iic"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_rightLim [TopologicalSpace α] [OrderTopology α] [T3Space β]
{f : α → β} {a : α} (h : Tendsto f (𝓝[>] a) (𝓝 (f.rightLim a))) :
f.rightLim.rightLim a = f.rightLim a | leftLim_leftLim (α := αᵒᵈ) h | theorem | rightLim_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"T3Space",
"TopologicalSpace",
"leftLim_leftLim"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim_rightLim [TopologicalSpace α] [OrderTopology α] [T3Space β]
{f : α → β} {a : α} (h : Tendsto f (𝓝[<] a) (𝓝 (f.leftLim a))) [h' : (𝓝[<] a).NeBot] :
f.rightLim.leftLim a = f.leftLim a | by
obtain ⟨b, hb⟩ : (Iio a).Nonempty := Filter.nonempty_of_mem (self_mem_nhdsWithin (a := a))
apply leftLim_eq_of_tendsto (neBot_iff.mp h')
apply (closed_nhds_basis (f.leftLim a)).tendsto_right_iff.2
rintro s ⟨s_mem, s_closed⟩
obtain ⟨u, au, hu⟩ : ∃ u, u < a ∧ Ioo u a ⊆ {x | f x ∈ s} := by
have := (closed... | theorem | leftLim_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"Filter.nonempty_of_mem",
"Ioo_mem_nhdsLT",
"OrderTopology",
"T3Space",
"TopologicalSpace",
"closed_nhds_basis",
"leftLim_eq_of_tendsto",
"mem_nhdsLT_iff_exists_Ioo_subset'",
"rightLim_eq_of_eq_bot",
"rightLim_eq_of_not_tendsto",
"self_mem_nhdsWithin",
"tendsto_rightLim_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_leftLim [TopologicalSpace α] [OrderTopology α] [T3Space β]
{f : α → β} {a : α} (h : Tendsto f (𝓝[>] a) (𝓝 (f.rightLim a))) [h' : (𝓝[>] a).NeBot] :
f.leftLim.rightLim a = f.rightLim a | leftLim_rightLim (α := αᵒᵈ) h (h' := h') | theorem | rightLim_leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"T3Space",
"TopologicalSpace",
"leftLim_rightLim"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop_of_mapClusterPt
[TopologicalSpace α] [OrderTopology α] [T3Space β] [NoTopOrder α] {f g : α → β} {b : β}
(h : Tendsto f atTop (𝓝 b)) (h' : ∀ᶠ x in atTop, MapClusterPt (g x) (𝓝 x) f) :
Tendsto g atTop (𝓝 b) | by
rcases isEmpty_or_nonempty α with hα | hα
· simp [filter_eq_bot_of_isEmpty atTop]
apply (closed_nhds_basis b).tendsto_right_iff.2
rintro s ⟨s_mem, s_closed⟩
obtain ⟨u, hu⟩ : ∃ a, ∀ (b : α), a ≤ b → MapClusterPt (g b) (𝓝 b) f ∧ f b ∈ s := by
simpa [eventually_atTop] using h'.and (h s_mem)
filter_upwa... | theorem | tendsto_atTop_of_mapClusterPt | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"Ici_mem_nhds",
"MapClusterPt",
"NoTopOrder",
"OrderTopology",
"T3Space",
"TopologicalSpace",
"closed_nhds_basis",
"isEmpty_or_nonempty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atBot_of_mapClusterPt
[TopologicalSpace α] [OrderTopology α] [T3Space β] [NoBotOrder α] {f g : α → β} {b : β}
(h : Tendsto f atBot (𝓝 b)) (h' : ∀ᶠ x in atBot, MapClusterPt (g x) (𝓝 x) f) :
Tendsto g atBot (𝓝 b) | tendsto_atTop_of_mapClusterPt (α := αᵒᵈ) h h' | theorem | tendsto_atBot_of_mapClusterPt | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"MapClusterPt",
"NoBotOrder",
"OrderTopology",
"T3Space",
"TopologicalSpace",
"tendsto_atTop_of_mapClusterPt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_leftLim_atTop_of_tendsto
[TopologicalSpace α] [OrderTopology α] [NoTopOrder α] [T3Space β]
{f : α → β} {b : β} (h : Tendsto f atTop (𝓝 b)) :
Tendsto f.leftLim atTop (𝓝 b) | by
apply tendsto_atTop_of_mapClusterPt h (Eventually.of_forall (fun x ↦ ?_))
exact MapClusterPt.mono (mapClusterPt_leftLim _ _) nhdsWithin_le_nhds | theorem | tendsto_leftLim_atTop_of_tendsto | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"MapClusterPt.mono",
"NoTopOrder",
"OrderTopology",
"T3Space",
"TopologicalSpace",
"mapClusterPt_leftLim",
"nhdsWithin_le_nhds",
"tendsto_atTop_of_mapClusterPt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_rightLim_atTop_of_tendsto [TopologicalSpace α] [OrderTopology α] [T3Space β]
{f : α → β} {b : β} (h : Tendsto f atTop (𝓝 b)) :
Tendsto f.rightLim atTop (𝓝 b) | by
cases topOrderOrNoTopOrder α
· simp only [OrderTop.atTop_eq α] at h ⊢
have : f.rightLim ⊤ = f ⊤ := rightLim_eq_of_isTop isTop_top
rw [tendsto_nhds_unique h (tendsto_pure_nhds f ⊤), ← this]
apply tendsto_pure_nhds
· apply tendsto_atTop_of_mapClusterPt h (Eventually.of_forall (fun x ↦ ?_))
exact ... | theorem | tendsto_rightLim_atTop_of_tendsto | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"MapClusterPt.mono",
"OrderTopology",
"T3Space",
"TopologicalSpace",
"isTop_top",
"mapClusterPt_rightLim",
"nhdsWithin_le_nhds",
"rightLim_eq_of_isTop",
"tendsto_atTop_of_mapClusterPt",
"tendsto_nhds_unique",
"tendsto_pure_nhds",
"topOrderOrNoTopOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_rightLim_atBot_of_tendsto
[TopologicalSpace α] [OrderTopology α] [NoBotOrder α] [T3Space β]
{f : α → β} {b : β} (h : Tendsto f atBot (𝓝 b)) :
Tendsto f.rightLim atBot (𝓝 b) | tendsto_leftLim_atTop_of_tendsto (α := αᵒᵈ) h | theorem | tendsto_rightLim_atBot_of_tendsto | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"NoBotOrder",
"OrderTopology",
"T3Space",
"TopologicalSpace",
"tendsto_leftLim_atTop_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_leftLim_atBot_of_tendsto [TopologicalSpace α] [OrderTopology α] [T3Space β]
{f : α → β} {b : β} (h : Tendsto f atBot (𝓝 b)) :
Tendsto f.leftLim atBot (𝓝 b) | tendsto_rightLim_atTop_of_tendsto (α := αᵒᵈ) h | theorem | tendsto_leftLim_atBot_of_tendsto | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"T3Space",
"TopologicalSpace",
"tendsto_rightLim_atTop_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim_eq_sSup [TopologicalSpace α] [OrderTopology α] (h : 𝓝[<] x ≠ ⊥) :
leftLim f x = sSup (f '' Iio x) | leftLim_eq_of_tendsto h (hf.tendsto_nhdsLT x) | theorem | Monotone.leftLim_eq_sSup | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"TopologicalSpace",
"leftLim_eq_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_eq_sInf [TopologicalSpace α] [OrderTopology α] (h : 𝓝[>] x ≠ ⊥) :
rightLim f x = sInf (f '' Ioi x) | rightLim_eq_of_tendsto h (hf.tendsto_nhdsGT x) | theorem | Monotone.rightLim_eq_sInf | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"TopologicalSpace",
"rightLim_eq_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim_le (h : x ≤ y) : leftLim f x ≤ f y | by
letI : TopologicalSpace α := Preorder.topology α
haveI : OrderTopology α := ⟨rfl⟩
rcases eq_or_ne (𝓝[<] x) ⊥ with (h' | h')
· simpa [leftLim, h'] using hf h
haveI A : NeBot (𝓝[<] x) := neBot_iff.2 h'
rw [leftLim_eq_sSup hf h']
refine csSup_le ?_ ?_
· simp only [image_nonempty]
exact (forall_mem... | theorem | Monotone.leftLim_le | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"Preorder.topology",
"TopologicalSpace",
"csSup_le",
"eq_or_ne",
"self_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_leftLim (h : x < y) : f x ≤ leftLim f y | by
letI : TopologicalSpace α := Preorder.topology α
haveI : OrderTopology α := ⟨rfl⟩
rcases eq_or_ne (𝓝[<] y) ⊥ with (h' | h')
· rw [leftLim_eq_of_eq_bot _ h']
exact hf h.le
rw [leftLim_eq_sSup hf h']
refine le_csSup ⟨f y, ?_⟩ (mem_image_of_mem _ h)
simp only [upperBounds, mem_image, mem_Iio, forall_... | theorem | Monotone.le_leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"OrderTopology",
"Preorder.topology",
"TopologicalSpace",
"eq_or_ne",
"le_csSup",
"leftLim_eq_of_eq_bot",
"upperBounds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim : Monotone (leftLim f) | by
intro x y h
rcases eq_or_lt_of_le h with (rfl | hxy)
· exact le_rfl
· exact (hf.leftLim_le le_rfl).trans (hf.le_leftLim hxy) | theorem | Monotone.leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"Monotone",
"eq_or_lt_of_le",
"le_rfl",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_rightLim (h : x ≤ y) : f x ≤ rightLim f y | hf.dual.leftLim_le h | theorem | Monotone.le_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_le (h : x < y) : rightLim f x ≤ f y | hf.dual.le_leftLim h | theorem | Monotone.rightLim_le | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim : Monotone (rightLim f) | fun _ _ h => hf.dual.leftLim h | theorem | Monotone.rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"Monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim_le_rightLim (h : x ≤ y) : leftLim f x ≤ rightLim f y | (hf.leftLim_le le_rfl).trans (hf.le_rightLim h) | theorem | Monotone.leftLim_le_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"le_rfl",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_le_leftLim (h : x < y) : rightLim f x ≤ leftLim f y | by
letI : TopologicalSpace α := Preorder.topology α
haveI : OrderTopology α := ⟨rfl⟩
rcases eq_or_neBot (𝓝[<] y) with (h' | h')
· simpa [leftLim, h'] using rightLim_le hf h
obtain ⟨a, ⟨xa, ay⟩⟩ : (Ioo x y).Nonempty := nonempty_of_mem (Ioo_mem_nhdsLT h)
calc
rightLim f x ≤ f a := hf.rightLim_le xa
_... | theorem | Monotone.rightLim_le_leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"Ioo_mem_nhdsLT",
"OrderTopology",
"Preorder.topology",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_leftLim (x : α) : Tendsto f (𝓝[<] x) (𝓝 (leftLim f x)) | tendsto_leftLim_of_tendsto ⟨_, hf.tendsto_nhdsLT x⟩ | theorem | Monotone.tendsto_leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"tendsto_leftLim_of_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≤] leftLim f x) | by
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within f (hf.tendsto_leftLim x)
filter_upwards [@self_mem_nhdsWithin _ _ x (Iio x)] with y hy using hf.le_leftLim hy | theorem | Monotone.tendsto_leftLim_within | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"self_mem_nhdsWithin",
"tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_rightLim (x : α) : Tendsto f (𝓝[>] x) (𝓝 (rightLim f x)) | hf.dual.tendsto_leftLim x | theorem | Monotone.tendsto_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_rightLim_within (x : α) : Tendsto f (𝓝[>] x) (𝓝[≥] rightLim f x) | hf.dual.tendsto_leftLim_within x | theorem | Monotone.tendsto_rightLim_within | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_Iio_iff_leftLim_eq :
ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x | by
rcases eq_or_ne (𝓝[<] x) ⊥ with (h' | h')
· simp [leftLim_eq_of_eq_bot f h', ContinuousWithinAt, h']
haveI : (𝓝[Iio x] x).NeBot := neBot_iff.2 h'
refine ⟨fun h => tendsto_nhds_unique (hf.tendsto_leftLim x) h.tendsto, fun h => ?_⟩
have := hf.tendsto_leftLim x
rwa [h] at this | theorem | Monotone.continuousWithinAt_Iio_iff_leftLim_eq | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ContinuousWithinAt",
"eq_or_ne",
"leftLim_eq_of_eq_bot",
"tendsto_nhds_unique"
] | A monotone function is continuous to the left at a point if and only if its left limit
coincides with the value of the function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_Ioi_iff_rightLim_eq :
ContinuousWithinAt f (Ioi x) x ↔ rightLim f x = f x | hf.dual.continuousWithinAt_Iio_iff_leftLim_eq | theorem | Monotone.continuousWithinAt_Ioi_iff_rightLim_eq | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ContinuousWithinAt"
] | A monotone function is continuous to the right at a point if and only if its right limit
coincides with the value of the function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAt_iff_leftLim_eq_rightLim : ContinuousAt f x ↔ leftLim f x = rightLim f x | by
refine ⟨fun h => ?_, fun h => ?_⟩
· have A : leftLim f x = f x :=
hf.continuousWithinAt_Iio_iff_leftLim_eq.1 h.continuousWithinAt
have B : rightLim f x = f x :=
hf.continuousWithinAt_Ioi_iff_rightLim_eq.1 h.continuousWithinAt
exact A.trans B.symm
· have h' : leftLim f x = f x := by
ap... | theorem | Monotone.continuousAt_iff_leftLim_eq_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ContinuousAt",
"le_antisymm",
"le_refl"
] | A monotone function is continuous at a point if and only if its left and right limits
coincide. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
le_leftLim (h : x ≤ y) : f y ≤ leftLim f x | hf.dual_right.leftLim_le h | theorem | Antitone.le_leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim_le (h : x < y) : leftLim f y ≤ f x | hf.dual_right.le_leftLim h | theorem | Antitone.leftLim_le | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim : Antitone (leftLim f) | hf.dual_right.leftLim | theorem | Antitone.leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"Antitone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_le (h : x ≤ y) : rightLim f y ≤ f x | hf.dual_right.le_rightLim h | theorem | Antitone.rightLim_le | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_rightLim (h : x < y) : f y ≤ rightLim f x | hf.dual_right.rightLim_le h | theorem | Antitone.le_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim : Antitone (rightLim f) | hf.dual_right.rightLim | theorem | Antitone.rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"Antitone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rightLim_le_leftLim (h : x ≤ y) : rightLim f y ≤ leftLim f x | hf.dual_right.leftLim_le_rightLim h | theorem | Antitone.rightLim_le_leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftLim_le_rightLim (h : x < y) : leftLim f y ≤ rightLim f x | hf.dual_right.rightLim_le_leftLim h | theorem | Antitone.leftLim_le_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_leftLim (x : α) : Tendsto f (𝓝[<] x) (𝓝 (leftLim f x)) | hf.dual_right.tendsto_leftLim x | theorem | Antitone.tendsto_leftLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≥] leftLim f x) | hf.dual_right.tendsto_leftLim_within x | theorem | Antitone.tendsto_leftLim_within | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_rightLim (x : α) : Tendsto f (𝓝[>] x) (𝓝 (rightLim f x)) | hf.dual_right.tendsto_rightLim x | theorem | Antitone.tendsto_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_rightLim_within (x : α) : Tendsto f (𝓝[>] x) (𝓝[≤] rightLim f x) | hf.dual_right.tendsto_rightLim_within x | theorem | Antitone.tendsto_rightLim_within | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_Iio_iff_leftLim_eq :
ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x | hf.dual_right.continuousWithinAt_Iio_iff_leftLim_eq | theorem | Antitone.continuousWithinAt_Iio_iff_leftLim_eq | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ContinuousWithinAt"
] | An antitone function is continuous to the left at a point if and only if its left limit
coincides with the value of the function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_Ioi_iff_rightLim_eq :
ContinuousWithinAt f (Ioi x) x ↔ rightLim f x = f x | hf.dual_right.continuousWithinAt_Ioi_iff_rightLim_eq | theorem | Antitone.continuousWithinAt_Ioi_iff_rightLim_eq | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ContinuousWithinAt"
] | An antitone function is continuous to the right at a point if and only if its right limit
coincides with the value of the function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAt_iff_leftLim_eq_rightLim : ContinuousAt f x ↔ leftLim f x = rightLim f x | hf.dual_right.continuousAt_iff_leftLim_eq_rightLim | theorem | Antitone.continuousAt_iff_leftLim_eq_rightLim | Topology.Order | Mathlib/Topology/Order/LeftRightLim.lean | [] | [
"ContinuousAt"
] | An antitone function is continuous at a point if and only if its left and right limits
coincide. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TFAE_mem_nhdsGT {a b : α} (hab : a < b) (s : Set α) :
TFAE [s ∈ 𝓝[>] a,
s ∈ 𝓝[Ioc a b] a,
s ∈ 𝓝[Ioo a b] a,
∃ u ∈ Ioc a b, Ioo a u ⊆ s,
∃ u ∈ Ioi a, Ioo a u ⊆ s] | by
tfae_have 1 ↔ 2 := by
rw [nhdsWithin_Ioc_eq_nhdsGT hab]
tfae_have 1 ↔ 3 := by
rw [nhdsWithin_Ioo_eq_nhdsGT hab]
tfae_have 4 → 5 := fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩
tfae_have 5 → 1
| ⟨u, hau, hu⟩ => mem_of_superset (Ioo_mem_nhdsGT hau) hu
tfae_have 1 → 4
| h => by
rcases mem_nhdsWithin_i... | theorem | TFAE_mem_nhdsGT | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set"
] | The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsGT_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :
s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s | (TFAE_mem_nhdsGT hu' s).out 0 3 | theorem | mem_nhdsGT_iff_exists_mem_Ioc_Ioo_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"TFAE_mem_nhdsGT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_nhdsGT_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :
s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s | (TFAE_mem_nhdsGT hu' s).out 0 4 | theorem | mem_nhdsGT_iff_exists_Ioo_subset' | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"TFAE_mem_nhdsGT"
] | A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`
with `a < u < u'`, provided `a` is not a top element. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhdsGT_basis_of_exists_gt {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) | let ⟨_, h⟩ := h
⟨fun _ => mem_nhdsGT_iff_exists_Ioo_subset' h⟩ | theorem | nhdsGT_basis_of_exists_gt | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"mem_nhdsGT_iff_exists_Ioo_subset'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsGT_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) | nhdsGT_basis_of_exists_gt <| exists_gt a | lemma | nhdsGT_basis | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMaxOrder",
"nhdsGT_basis_of_exists_gt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsGT_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b | by
by_cases ha : IsTop a
· simp [ha, ha.isMax.Ioi_eq]
· simp only [ha, false_or]
rw [isTop_iff_isMax, not_isMax_iff] at ha
simp only [(nhdsGT_basis_of_exists_gt ha).eq_bot_iff, covBy_iff_Ioo_eq] | theorem | nhdsGT_eq_bot_iff | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"by_cases",
"covBy_iff_Ioo_eq",
"isTop_iff_isMax",
"nhdsGT_basis_of_exists_gt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_nhdsGT_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :
s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s | let ⟨_u', hu'⟩ := exists_gt a
mem_nhdsGT_iff_exists_Ioo_subset' hu' | theorem | mem_nhdsGT_iff_exists_Ioo_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMaxOrder",
"Set",
"mem_nhdsGT_iff_exists_Ioo_subset'"
] | A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`
with `a < u`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
countable_setOf_isolated_right [SecondCountableTopology α] :
{ x : α | 𝓝[>] x = ⊥ }.Countable | by
simp only [nhdsGT_eq_bot_iff, setOf_or]
exact (subsingleton_isTop α).countable.union countable_setOf_covBy_right | theorem | countable_setOf_isolated_right | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Countable",
"SecondCountableTopology",
"countable_setOf_covBy_right",
"nhdsGT_eq_bot_iff"
] | The set of points which are isolated on the right is countable when the space is
second-countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
countable_setOf_isolated_left [SecondCountableTopology α] :
{ x : α | 𝓝[<] x = ⊥ }.Countable | countable_setOf_isolated_right (α := αᵒᵈ) | theorem | countable_setOf_isolated_left | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Countable",
"SecondCountableTopology",
"countable_setOf_isolated_right"
] | The set of points which are isolated on the left is countable when the space is
second-countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
countable_setOf_isolated_right_within [SecondCountableTopology α] {s : Set α} :
{ x ∈ s | 𝓝[s ∩ Ioi x] x = ⊥ }.Countable | by
/- This does not follow from `countable_setOf_isolated_right`, which gives the result when `s`
is the whole space, as one cannot use it inside the subspace since it doesn't have the order
topology. Instead, we follow the main steps of its proof. -/
let t := { x ∈ s | 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬ IsTop x}
suffic... | theorem | countable_setOf_isolated_right_within | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Countable",
"Function.onFun",
"SecondCountableTopology",
"Set",
"Set.Pairwise",
"Set.PairwiseDisjoint.countable_of_Ioo",
"by_cases",
"lt_of_le_of_ne",
"mem_nhdsWithin_iff_exists_mem_nhds_inter",
"not_le",
"symm",
"trans"
] | The set of points in a set which are isolated on the right in this set is countable when the
space is second-countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
countable_setOf_isolated_left_within [SecondCountableTopology α] {s : Set α} :
{ x ∈ s | 𝓝[s ∩ Iio x] x = ⊥ }.Countable | countable_setOf_isolated_right_within (α := αᵒᵈ) | theorem | countable_setOf_isolated_left_within | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Countable",
"SecondCountableTopology",
"Set",
"countable_setOf_isolated_right_within"
] | The set of points in a set which are isolated on the left in this set is countable when the
space is second-countable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsGT_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α} {s : Set α} :
s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s | by
rw [mem_nhdsGT_iff_exists_Ioo_subset]
constructor
· rintro ⟨u, au, as⟩
rcases exists_between au with ⟨v, hv⟩
exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩
· rintro ⟨u, au, as⟩
exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩ | theorem | mem_nhdsGT_iff_exists_Ioc_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"DenselyOrdered",
"NoMaxOrder",
"Set",
"exists_between",
"lt_of_le_of_lt",
"mem_nhdsGT_iff_exists_Ioo_subset"
] | A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TFAE_mem_nhdsLT {a b : α} (h : a < b) (s : Set α) :
TFAE [s ∈ 𝓝[<] b, -- 0 : `s` is a neighborhood of `b` within `(-∞, b)`
s ∈ 𝓝[Ico a b] b, -- 1 : `s` is a neighborhood of `b` within `[a, b)`
s ∈ 𝓝[Ioo a b] b, -- 2 : `s` is a neighborhood of `b` within `(a, b)`
∃ l ∈ Ico a b, Ioo l b ⊆ s... | by -- 4 : `s` includes `(l, b)` for some `l < b`
simpa using! TFAE_mem_nhdsGT h.dual (ofDual ⁻¹' s) | theorem | TFAE_mem_nhdsLT | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"TFAE_mem_nhdsGT"
] | The following statements are equivalent:
0. `s` is a neighborhood of `b` within `(-∞, b)`
1. `s` is a neighborhood of `b` within `[a, b)`
2. `s` is a neighborhood of `b` within `(a, b)`
3. `s` includes `(l, b)` for some `l ∈ [a, b)`
4. `s` includes `(l, b)` for some `l < b` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsLT_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :
s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s | (TFAE_mem_nhdsLT hl' s).out 0 3 | theorem | mem_nhdsLT_iff_exists_mem_Ico_Ioo_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"TFAE_mem_nhdsLT"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_nhdsLT_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :
s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s | (TFAE_mem_nhdsLT hl' s).out 0 4 | theorem | mem_nhdsLT_iff_exists_Ioo_subset' | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"TFAE_mem_nhdsLT"
] | A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`
with `l < a`, provided `a` is not a bottom element. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsLT_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :
s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s | let ⟨_, h⟩ := exists_lt a
mem_nhdsLT_iff_exists_Ioo_subset' h | theorem | mem_nhdsLT_iff_exists_Ioo_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMinOrder",
"Set",
"mem_nhdsLT_iff_exists_Ioo_subset'"
] | A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`
with `l < a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsLT_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α} {s : Set α} :
s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s | by
have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsGT_iff_exists_Ioc_subset
simpa using! this | theorem | mem_nhdsLT_iff_exists_Ico_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"DenselyOrdered",
"NoMinOrder",
"Set",
"mem_nhdsGT_iff_exists_Ioc_subset"
] | A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`
with `l < a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhdsLT_basis_of_exists_lt {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) | let ⟨_, h⟩ := h
⟨fun _ => mem_nhdsLT_iff_exists_Ioo_subset' h⟩ | theorem | nhdsLT_basis_of_exists_lt | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"mem_nhdsLT_iff_exists_Ioo_subset'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsLT_basis [NoMinOrder α] (a : α) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) | nhdsLT_basis_of_exists_lt <| exists_lt a | theorem | nhdsLT_basis | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMinOrder",
"nhdsLT_basis_of_exists_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsLT_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a | by
convert! (config := { preTransparency := .default })
nhdsGT_eq_bot_iff (a := OrderDual.toDual a) using 4
exact ofDual_covBy_ofDual_iff | theorem | nhdsLT_eq_bot_iff | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"IsBot",
"OrderDual.toDual",
"nhdsGT_eq_bot_iff",
"ofDual_covBy_ofDual_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TFAE_mem_nhdsGE {a b : α} (hab : a < b) (s : Set α) :
TFAE [s ∈ 𝓝[≥] a,
s ∈ 𝓝[Icc a b] a,
s ∈ 𝓝[Ico a b] a,
∃ u ∈ Ioc a b, Ico a u ⊆ s,
∃ u ∈ Ioi a, Ico a u ⊆ s] | by
tfae_have 1 ↔ 2 := by
rw [nhdsWithin_Icc_eq_nhdsGE hab]
tfae_have 1 ↔ 3 := by
rw [nhdsWithin_Ico_eq_nhdsGE hab]
tfae_have 1 ↔ 5 := (nhdsGE_basis_of_exists_gt ⟨b, hab⟩).mem_iff
tfae_have 4 → 5 := fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩
tfae_have 5 → 4
| ⟨u, hua, hus⟩ => ⟨min u b, ⟨lt_min hua hab, min... | theorem | TFAE_mem_nhdsGE | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"min_le_left",
"min_le_right",
"nhdsGE_basis_of_exists_gt",
"trans"
] | The following statements are equivalent:
0. `s` is a neighborhood of `a` within `[a, +∞)`;
1. `s` is a neighborhood of `a` within `[a, b]`;
2. `s` is a neighborhood of `a` within `[a, b)`;
3. `s` includes `[a, u)` for some `u ∈ (a, b]`;
4. `s` includes `[a, u)` for some `u > a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsGE_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :
s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s | (TFAE_mem_nhdsGE hu' s).out 0 3 (by simp) (by simp) | theorem | mem_nhdsGE_iff_exists_mem_Ioc_Ico_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"TFAE_mem_nhdsGE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_nhdsGE_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :
s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s | (TFAE_mem_nhdsGE hu' s).out 0 4 (by simp) (by simp) | theorem | mem_nhdsGE_iff_exists_Ico_subset' | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"TFAE_mem_nhdsGE"
] | A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`
with `a < u < u'`, provided `a` is not a top element. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsGE_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :
s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s | let ⟨_, hu'⟩ := exists_gt a
mem_nhdsGE_iff_exists_Ico_subset' hu' | theorem | mem_nhdsGE_iff_exists_Ico_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMaxOrder",
"Set",
"mem_nhdsGE_iff_exists_Ico_subset'"
] | A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`
with `a < u`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhdsGE_basis_Ico [NoMaxOrder α] (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) | ⟨fun _ => mem_nhdsGE_iff_exists_Ico_subset⟩ | theorem | nhdsGE_basis_Ico | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMaxOrder"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsGE_basis_Icc [NoMaxOrder α] [DenselyOrdered α] {a : α} :
(𝓝[≥] a).HasBasis (a < ·) (Icc a) | (nhdsGE_basis _).to_hasBasis
(fun _u hu ↦ (exists_between hu).imp fun _v hv ↦ hv.imp_right Icc_subset_Ico_right) fun u hu ↦
⟨u, hu, Ico_subset_Icc_self⟩ | theorem | nhdsGE_basis_Icc | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"DenselyOrdered",
"NoMaxOrder",
"exists_between",
"nhdsGE_basis"
] | The filter of right neighborhoods has a basis of closed intervals. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsGE_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α} {s : Set α} :
s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s | nhdsGE_basis_Icc.mem_iff | theorem | mem_nhdsGE_iff_exists_Icc_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"DenselyOrdered",
"NoMaxOrder",
"Set"
] | A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
with `a < u`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TFAE_mem_nhdsLE {a b : α} (h : a < b) (s : Set α) :
TFAE [s ∈ 𝓝[≤] b, -- 0 : `s` is a neighborhood of `b` within `(-∞, b]`
s ∈ 𝓝[Icc a b] b, -- 1 : `s` is a neighborhood of `b` within `[a, b]`
s ∈ 𝓝[Ioc a b] b, -- 2 : `s` is a neighborhood of `b` within `(a, b]`
∃ l ∈ Ico a b, Ioc l b ⊆ s, -- 3... | by -- 4 : `s` includes `(l, b]` for some `l < b`
simpa using! TFAE_mem_nhdsGE h.dual (ofDual ⁻¹' s) | theorem | TFAE_mem_nhdsLE | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"TFAE_mem_nhdsGE"
] | The following statements are equivalent:
0. `s` is a neighborhood of `b` within `(-∞, b]`
1. `s` is a neighborhood of `b` within `[a, b]`
2. `s` is a neighborhood of `b` within `(a, b]`
3. `s` includes `(l, b]` for some `l ∈ [a, b)`
4. `s` includes `(l, b]` for some `l < b` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsLE_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :
s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s | (TFAE_mem_nhdsLE hl' s).out 0 3 (by simp) (by simp) | theorem | mem_nhdsLE_iff_exists_mem_Ico_Ioc_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"TFAE_mem_nhdsLE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_nhdsLE_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :
s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s | (TFAE_mem_nhdsLE hl' s).out 0 4 (by simp) (by simp) | theorem | mem_nhdsLE_iff_exists_Ioc_subset' | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Set",
"TFAE_mem_nhdsLE"
] | A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`
with `l < a`, provided `a` is not a bottom element. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsLE_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :
s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s | let ⟨_, hl'⟩ := exists_lt a
mem_nhdsLE_iff_exists_Ioc_subset' hl' | theorem | mem_nhdsLE_iff_exists_Ioc_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMinOrder",
"Set",
"mem_nhdsLE_iff_exists_Ioc_subset'"
] | A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`
with `l < a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsLE_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s | calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl
_ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=
mem_nhdsGE_iff_exists_Icc_subset
_ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp | theorem | mem_nhdsLE_iff_exists_Icc_subset | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"DenselyOrdered",
"NoMinOrder",
"Set",
"mem_nhdsGE_iff_exists_Icc_subset"
] | A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`
with `l < a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhdsLE_basis_Icc [NoMinOrder α] [DenselyOrdered α] {a : α} :
(𝓝[≤] a).HasBasis (· < a) (Icc · a) | ⟨fun _ ↦ mem_nhdsLE_iff_exists_Icc_subset⟩ | theorem | nhdsLE_basis_Icc | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"DenselyOrdered",
"NoMinOrder"
] | The filter of left neighborhoods has a basis of closed intervals. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhds_eq_iInf_mabs_div (a : α) : 𝓝 a = ⨅ r > 1, 𝓟 { b | |a / b|ₘ < r } | by
simp only [nhds_eq_order, mabs_lt, setOf_and, ← inf_principal, iInf_inf_eq]
refine (congr_arg₂ _ ?_ ?_).trans (inf_comm ..)
· refine (Equiv.divLeft a).iInf_congr fun x => ?_; simp [Ioi]
· refine (Equiv.divRight a).iInf_congr fun x => ?_; simp [Iio] | theorem | nhds_eq_iInf_mabs_div | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Equiv.divLeft",
"Equiv.divRight",
"mabs_lt",
"nhds_eq_order",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
orderTopology_of_nhds_mabs {α : Type*} [TopologicalSpace α] [CommGroup α] [LinearOrder α]
[IsOrderedMonoid α]
(h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 1, 𝓟 { b | |a / b|ₘ < r }) : OrderTopology α | by
refine ⟨TopologicalSpace.ext_nhds fun a => ?_⟩
rw [h_nhds]
letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩
exact (nhds_eq_iInf_mabs_div a).symm | theorem | orderTopology_of_nhds_mabs | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"CommGroup",
"IsOrderedMonoid",
"LinearOrder",
"OrderTopology",
"Preorder.topology",
"TopologicalSpace",
"nhds_eq_iInf_mabs_div",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LinearOrderedCommGroup.tendsto_nhds {x : Filter β} {a : α} :
Tendsto f x (𝓝 a) ↔ ∀ ε > (1 : α), ∀ᶠ b in x, |f b / a|ₘ < ε | by
simp [nhds_eq_iInf_mabs_div, mabs_div_comm a] | theorem | LinearOrderedCommGroup.tendsto_nhds | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Filter",
"mabs_div_comm",
"nhds_eq_iInf_mabs_div"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_mabs_div_lt (a : α) {ε : α} (hε : 1 < ε) : ∀ᶠ x in 𝓝 a, |x / a|ₘ < ε | (nhds_eq_iInf_mabs_div a).symm ▸
mem_iInf_of_mem ε (mem_iInf_of_mem hε <| by simp only [mabs_div_comm, mem_principal_self]) | theorem | eventually_mabs_div_lt | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"mabs_div_comm",
"nhds_eq_iInf_mabs_div",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.mul_atTop' {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :
Tendsto (fun x => f x * g x) l atTop | by
nontriviality α
obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C
refine tendsto_atTop_mul_left_of_le' _ C' ?_ hg
exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt | theorem | Filter.Tendsto.mul_atTop' | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"le_of_lt",
"lt_mem_nhds"
] | In a linearly ordered commutative group with the order topology,
if `f` tends to `C` and `g` tends to `atTop` then `f * g` tends to `atTop`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.mul_atBot' {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :
Tendsto (fun x => f x * g x) l atBot | Filter.Tendsto.mul_atTop' (α := αᵒᵈ) hf hg | theorem | Filter.Tendsto.mul_atBot' | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"Filter.Tendsto.mul_atTop'"
] | In a linearly ordered commutative group with the order topology,
if `f` tends to `C` and `g` tends to `atBot` then `f * g` tends to `atBot`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.atTop_mul' {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :
Tendsto (fun x => f x * g x) l atTop | by
conv in _ * _ => rw [mul_comm]
exact hg.mul_atTop' hf | theorem | Filter.Tendsto.atTop_mul' | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"mul_comm"
] | In a linearly ordered commutative group with the order topology,
if `f` tends to `atTop` and `g` tends to `C` then `f * g` tends to `atTop`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.atBot_mul' {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :
Tendsto (fun x => f x * g x) l atBot | by
conv in _ * _ => rw [mul_comm]
exact hg.mul_atBot' hf | theorem | Filter.Tendsto.atBot_mul' | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"mul_comm"
] | In a linearly ordered commutative group with the order topology,
if `f` tends to `atBot` and `g` tends to `C` then `f * g` tends to `atBot`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhds_basis_mabs_div_lt [NoMaxOrder α] (a : α) :
(𝓝 a).HasBasis (fun ε : α => (1 : α) < ε) fun ε => { b | |b / a|ₘ < ε } | by
simp only [nhds_eq_iInf_mabs_div, mabs_div_comm (a := a)]
refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _)
exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _),
fun _ hz => hz.trans_le (min_le_right _ _)⟩ | theorem | nhds_basis_mabs_div_lt | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMaxOrder",
"lt_min",
"mabs_div_comm",
"min_le_left",
"min_le_right",
"nhds_eq_iInf_mabs_div"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_Ioo_one_lt [NoMaxOrder α] (a : α) :
(𝓝 a).HasBasis (fun ε : α => (1 : α) < ε) fun ε => Ioo (a / ε) (a * ε) | by
convert! nhds_basis_mabs_div_lt a
simp only [Ioo, mabs_lt, ← div_lt_iff_lt_mul, inv_lt_div_iff_lt_mul, div_lt_comm] | theorem | nhds_basis_Ioo_one_lt | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMaxOrder",
"div_lt_comm",
"div_lt_iff_lt_mul",
"inv_lt_div_iff_lt_mul",
"mabs_lt",
"nhds_basis_mabs_div_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_Icc_one_lt [NoMaxOrder α] [DenselyOrdered α] (a : α) :
(𝓝 a).HasBasis ((1 : α) < ·) fun ε ↦ Icc (a / ε) (a * ε) | (nhds_basis_Ioo_one_lt a).to_hasBasis
(fun _ε ε₁ ↦ let ⟨δ, δ₁, δε⟩ := exists_between ε₁
⟨δ, δ₁, Icc_subset_Ioo (by gcongr) (by gcongr)⟩)
(fun ε ε₁ ↦ ⟨ε, ε₁, Ioo_subset_Icc_self⟩) | theorem | nhds_basis_Icc_one_lt | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"DenselyOrdered",
"NoMaxOrder",
"exists_between",
"nhds_basis_Ioo_one_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_one_mabs_lt [NoMaxOrder α] :
(𝓝 (1 : α)).HasBasis (fun ε : α => (1 : α) < ε) fun ε => { b | |b|ₘ < ε } | by
simpa using nhds_basis_mabs_div_lt (1 : α) | theorem | nhds_basis_one_mabs_lt | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMaxOrder",
"nhds_basis_mabs_div_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_Ioo_one_lt_of_one_lt [NoMaxOrder α] {a : α} (ha : 1 < a) :
(𝓝 a).HasBasis (fun ε : α => (1 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a / ε) (a * ε) | (nhds_basis_Ioo_one_lt a).restrict fun ε hε ↦
⟨min a ε, lt_min ha hε, min_le_left _ _, by gcongr <;> apply min_le_right⟩ | theorem | nhds_basis_Ioo_one_lt_of_one_lt | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"NoMaxOrder",
"lt_min",
"min_le_left",
"nhds_basis_Ioo_one_lt"
] | If `a > 1`, then open intervals `(a / ε, aε)`, `1 < ε ≤ a`,
form a basis of neighborhoods of `a`.
This upper bound for `ε` guarantees that all elements of these intervals are greater than one. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsGE (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[≥] x | mem_of_superset (Icc_mem_nhdsGE hxy) <| hS.out hx hy | lemma | Set.OrdConnected.mem_nhdsGE | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [] | If `S` is order-connected and contains two points `x < y`,
then `S` is a right neighbourhood of `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsGT (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[>] x | nhdsWithin_mono _ Ioi_subset_Ici_self <| hS.mem_nhdsGE hx hy hxy | lemma | Set.OrdConnected.mem_nhdsGT | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [
"nhdsWithin_mono"
] | If `S` is order-connected and contains two points `x < y`,
then `S` is a punctured right neighbourhood of `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsLE (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[≤] y | hS.dual.mem_nhdsGE hy hx hxy | lemma | Set.OrdConnected.mem_nhdsLE | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [] | If `S` is order-connected and contains two points `x < y`, then `S` is a left neighbourhood
of `y`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_nhdsLT (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[<] y | hS.dual.mem_nhdsGT hy hx hxy | lemma | Set.OrdConnected.mem_nhdsLT | Topology.Order | Mathlib/Topology/Order/LeftRightNhds.lean | [] | [] | If `S` is order-connected and contains two points `x < y`, then `S` is a punctured left
neighbourhood of `y`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
BoundedLENhdsClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
isBounded_le_nhds (a : α) : (𝓝 a).IsBounded (· ≤ ·) | class | BoundedLENhdsClass | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Preorder",
"TopologicalSpace",
"isBounded_le_nhds"
] | Ad hoc typeclass stating that neighborhoods are eventually bounded above. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
BoundedGENhdsClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
isBounded_ge_nhds (a : α) : (𝓝 a).IsBounded (· ≥ ·) | class | BoundedGENhdsClass | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [
"Preorder",
"TopologicalSpace",
"isBounded_ge_nhds"
] | Ad hoc typeclass stating that neighborhoods are eventually bounded below. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isBounded_le_nhds (a : α) : (𝓝 a).IsBounded (· ≤ ·) | BoundedLENhdsClass.isBounded_le_nhds _ | theorem | isBounded_le_nhds | Topology.Order | Mathlib/Topology/Order/LiminfLimsup.lean | [
"Mathlib.Data.Fintype.Order",
"Mathlib.Topology.Order.MonotoneConvergence"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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