fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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continuousWithinAt_Ioi_iff_Ici {a : α} {f : α → β} :
ContinuousWithinAt f (Ioi a) a ↔ ContinuousWithinAt f (Ici a) a := by
simp only [← Ici_diff_left, continuousWithinAt_diff_self] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | continuousWithinAt_Ioi_iff_Ici | null |
continuousWithinAt_Iio_iff_Iic {a : α} {f : α → β} :
ContinuousWithinAt f (Iio a) a ↔ ContinuousWithinAt f (Iic a) a :=
continuousWithinAt_Ioi_iff_Ici (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | continuousWithinAt_Iio_iff_Iic | null |
continuousWithinAt_inter_Ioi_iff_Ici {a : α} {f : α → β} {s : Set α} :
ContinuousWithinAt f (s ∩ Ioi a) a ↔ ContinuousWithinAt f (s ∩ Ici a) a := by
simp [← Ici_diff_left, ← inter_diff_assoc, continuousWithinAt_diff_self] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | continuousWithinAt_inter_Ioi_iff_Ici | null |
continuousWithinAt_inter_Iio_iff_Iic {a : α} {f : α → β} {s : Set α} :
ContinuousWithinAt f (s ∩ Iio a) a ↔ ContinuousWithinAt f (s ∩ Iic a) a :=
continuousWithinAt_inter_Ioi_iff_Ici (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | continuousWithinAt_inter_Iio_iff_Iic | null |
nhdsLE_sup_nhdsGE (a : α) : 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a := by
rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsLE_sup_nhdsGE | null |
nhdsWithinLE_sup_nhdsWithinGE (a : α) : 𝓝[s ∩ Iic a] a ⊔ 𝓝[s ∩ Ici a] a = 𝓝[s] a := by
rw [← nhdsWithin_union, ← inter_union_distrib_left, Iic_union_Ici, inter_univ] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsWithinLE_sup_nhdsWithinGE | null |
nhdsLT_sup_nhdsGE (a : α) : 𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a := by
rw [← nhdsWithin_union, Iio_union_Ici, nhdsWithin_univ] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsLT_sup_nhdsGE | null |
nhdsWithinLT_sup_nhdsWithinGE (a : α) : 𝓝[s ∩ Iio a] a ⊔ 𝓝[s ∩ Ici a] a = 𝓝[s] a := by
rw [← nhdsWithin_union, ← inter_union_distrib_left, Iio_union_Ici, inter_univ] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsWithinLT_sup_nhdsWithinGE | null |
nhdsLE_sup_nhdsGT (a : α) : 𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a := by
rw [← nhdsWithin_union, Iic_union_Ioi, nhdsWithin_univ] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsLE_sup_nhdsGT | null |
nhdsWithinLE_sup_nhdsWithinGT (a : α) : 𝓝[s ∩ Iic a] a ⊔ 𝓝[s ∩ Ioi a] a = 𝓝[s] a := by
rw [← nhdsWithin_union, ← inter_union_distrib_left, Iic_union_Ioi, inter_univ] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsWithinLE_sup_nhdsWithinGT | null |
nhdsLT_sup_nhdsGT (a : α) : 𝓝[<] a ⊔ 𝓝[>] a = 𝓝[≠] a := by
rw [← nhdsWithin_union, Iio_union_Ioi] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsLT_sup_nhdsGT | null |
nhdsWithinLT_sup_nhdsWithinGT (a : α) :
𝓝[s ∩ Iio a] a ⊔ 𝓝[s ∩ Ioi a] a = 𝓝[s \ {a}] a := by
rw [← nhdsWithin_union, ← inter_union_distrib_left, Iio_union_Ioi, compl_eq_univ_diff,
inter_sdiff_left_comm, univ_inter] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsWithinLT_sup_nhdsWithinGT | null |
nhdsGT_sup_nhdsWithin_singleton (a : α) :
𝓝[>] a ⊔ 𝓝[{a}] a = 𝓝[≥] a := by
simp only [union_singleton, Ioi_insert, ← nhdsWithin_union]
@[deprecated (since := "2025-06-15")]
alias nhdsWithin_right_sup_nhds_singleton := nhdsGT_sup_nhdsWithin_singleton | lemma | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | nhdsGT_sup_nhdsWithin_singleton | null |
continuousAt_iff_continuous_left_right {a : α} {f : α → β} :
ContinuousAt f a ↔ ContinuousWithinAt f (Iic a) a ∧ ContinuousWithinAt f (Ici a) a := by
simp only [ContinuousWithinAt, ContinuousAt, ← tendsto_sup, nhdsLE_sup_nhdsGE] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | continuousAt_iff_continuous_left_right | null |
continuousAt_iff_continuous_left'_right' {a : α} {f : α → β} :
ContinuousAt f a ↔ ContinuousWithinAt f (Iio a) a ∧ ContinuousWithinAt f (Ioi a) a := by
rw [continuousWithinAt_Ioi_iff_Ici, continuousWithinAt_Iio_iff_Iic,
continuousAt_iff_continuous_left_right] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | continuousAt_iff_continuous_left'_right' | null |
continuousWithinAt_iff_continuous_left_right {a : α} {f : α → β} :
ContinuousWithinAt f s a ↔
ContinuousWithinAt f (s ∩ Iic a) a ∧ ContinuousWithinAt f (s ∩ Ici a) a := by
simp only [ContinuousWithinAt, ← tendsto_sup, nhdsWithinLE_sup_nhdsWithinGE] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | continuousWithinAt_iff_continuous_left_right | null |
continuousWithinAt_iff_continuous_left'_right' {a : α} {f : α → β} :
ContinuousWithinAt f s a ↔
ContinuousWithinAt f (s ∩ Iio a) a ∧ ContinuousWithinAt f (s ∩ Ioi a) a := by
rw [continuousWithinAt_inter_Ioi_iff_Ici, continuousWithinAt_inter_Iio_iff_Iic,
continuousWithinAt_iff_continuous_left_right] | theorem | Topology | [
"Mathlib.Order.Antichain",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/Order/LeftRight.lean | continuousWithinAt_iff_continuous_left'_right' | null |
noncomputable Function.leftLim (f : α → β) (a : α) : β := by
classical
haveI : Nonempty β := ⟨f a⟩
letI : TopologicalSpace α := Preorder.topology α
exact if 𝓝[<] a = ⊥ ∨ ¬∃ y, Tendsto f (𝓝[<] a) (𝓝 y) then f a else limUnder (𝓝[<] a) f | def | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | Function.leftLim | Let `f : α → β` be a function from a linear order `α` to a topological space `β`, and
let `a : α`. The limit strictly to the left of `f` at `a`, denoted with `leftLim f a`, is defined
by using the order topology on `α`. If `a` is isolated to its left or the function has no left
limit, we use `f a` instead to guarantee a good behavior in most cases. |
noncomputable Function.rightLim (f : α → β) (a : α) : β :=
@Function.leftLim αᵒᵈ β _ _ f a
open Function | def | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | Function.rightLim | Let `f : α → β` be a function from a linear order `α` to a topological space `β`, and
let `a : α`. The limit strictly to the right of `f` at `a`, denoted with `rightLim f a`, is defined
by using the order topology on `α`. If `a` is isolated to its right or the function has no right
limit, we use `f a` instead to guarantee a good behavior in most cases. |
leftLim_eq_of_tendsto [hα : TopologicalSpace α] [h'α : OrderTopology α] [T2Space β]
{f : α → β} {a : α} {y : β} (h : 𝓝[<] a ≠ ⊥) (h' : Tendsto f (𝓝[<] a) (𝓝 y)) :
leftLim f a = y := by
have h'' : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y) := ⟨y, h'⟩
rw [h'α.topology_eq_generate_intervals] at h h' h''
simp only [leftLim, h, h'', not_true, or_self_iff, if_false]
haveI := neBot_iff.2 h
exact lim_eq h' | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | leftLim_eq_of_tendsto | null |
leftLim_eq_of_eq_bot [hα : TopologicalSpace α] [h'α : OrderTopology α] (f : α → β) {a : α}
(h : 𝓝[<] a = ⊥) : leftLim f a = f a := by
rw [h'α.topology_eq_generate_intervals] at h
simp [leftLim, h] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | leftLim_eq_of_eq_bot | null |
rightLim_eq_of_tendsto [TopologicalSpace α] [OrderTopology α] [T2Space β]
{f : α → β} {a : α} {y : β} (h : 𝓝[>] a ≠ ⊥) (h' : Tendsto f (𝓝[>] a) (𝓝 y)) :
Function.rightLim f a = y :=
@leftLim_eq_of_tendsto αᵒᵈ _ _ _ _ _ _ f a y h h' | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | rightLim_eq_of_tendsto | null |
rightLim_eq_of_eq_bot [TopologicalSpace α] [OrderTopology α] (f : α → β) {a : α}
(h : 𝓝[>] a = ⊥) : rightLim f a = f a :=
@leftLim_eq_of_eq_bot αᵒᵈ _ _ _ _ _ f a h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | rightLim_eq_of_eq_bot | null |
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 | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | leftLim_eq_sSup | null |
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 | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | rightLim_eq_sInf | null |
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_nonempty_iff_neBot.2 A) _ self_mem_nhdsWithin
· simp only [mem_image, mem_Iio, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
intro z hz
exact hf (hz.le.trans h) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | leftLim_le | null |
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_exists_index, and_imp,
forall_apply_eq_imp_iff₂, mem_setOf_eq]
intro z hz
exact hf hz.le
@[mono] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | le_leftLim | null |
protected 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 | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | leftLim | null |
le_rightLim (h : x ≤ y) : f x ≤ rightLim f y :=
hf.dual.leftLim_le h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | le_rightLim | null |
rightLim_le (h : x < y) : rightLim f x ≤ f y :=
hf.dual.le_leftLim h
@[mono] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | rightLim_le | null |
protected rightLim : Monotone (rightLim f) := fun _ _ h => hf.dual.leftLim h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | rightLim | null |
leftLim_le_rightLim (h : x ≤ y) : leftLim f x ≤ rightLim f y :=
(hf.leftLim_le le_rfl).trans (hf.le_rightLim h) | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | leftLim_le_rightLim | null |
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
_ ≤ leftLim f y := hf.le_leftLim ay
variable [TopologicalSpace α] [OrderTopology α] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | rightLim_le_leftLim | null |
tendsto_leftLim (x : α) : Tendsto f (𝓝[<] x) (𝓝 (leftLim f x)) := by
rcases eq_or_ne (𝓝[<] x) ⊥ with (h' | h')
· simp [h']
rw [leftLim_eq_sSup hf h']
exact hf.tendsto_nhdsLT x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | tendsto_leftLim | null |
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 | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | tendsto_leftLim_within | null |
tendsto_rightLim (x : α) : Tendsto f (𝓝[>] x) (𝓝 (rightLim f x)) :=
hf.dual.tendsto_leftLim x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | tendsto_rightLim | null |
tendsto_rightLim_within (x : α) : Tendsto f (𝓝[>] x) (𝓝[≥] rightLim f x) :=
hf.dual.tendsto_leftLim_within x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | tendsto_rightLim_within | null |
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 | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | continuousWithinAt_Iio_iff_leftLim_eq | 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. |
continuousWithinAt_Ioi_iff_rightLim_eq :
ContinuousWithinAt f (Ioi x) x ↔ rightLim f x = f x :=
hf.dual.continuousWithinAt_Iio_iff_leftLim_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | continuousWithinAt_Ioi_iff_rightLim_eq | 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. |
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
apply le_antisymm (leftLim_le hf (le_refl _))
rw [h]
exact le_rightLim hf (le_refl _)
refine continuousAt_iff_continuous_left'_right'.2 ⟨?_, ?_⟩
· exact hf.continuousWithinAt_Iio_iff_leftLim_eq.2 h'
· rw [h] at h'
exact hf.continuousWithinAt_Ioi_iff_rightLim_eq.2 h' | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | continuousAt_iff_leftLim_eq_rightLim | A monotone function is continuous at a point if and only if its left and right limits
coincide. |
le_leftLim (h : x ≤ y) : f y ≤ leftLim f x :=
hf.dual_right.leftLim_le h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | le_leftLim | null |
leftLim_le (h : x < y) : leftLim f y ≤ f x :=
hf.dual_right.le_leftLim h
@[mono] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | leftLim_le | null |
protected leftLim : Antitone (leftLim f) :=
hf.dual_right.leftLim | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | leftLim | null |
rightLim_le (h : x ≤ y) : rightLim f y ≤ f x :=
hf.dual_right.le_rightLim h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | rightLim_le | null |
le_rightLim (h : x < y) : f y ≤ rightLim f x :=
hf.dual_right.rightLim_le h
@[mono] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | le_rightLim | null |
protected rightLim : Antitone (rightLim f) :=
hf.dual_right.rightLim | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | rightLim | null |
rightLim_le_leftLim (h : x ≤ y) : rightLim f y ≤ leftLim f x :=
hf.dual_right.leftLim_le_rightLim h | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | rightLim_le_leftLim | null |
leftLim_le_rightLim (h : x < y) : leftLim f y ≤ rightLim f x :=
hf.dual_right.rightLim_le_leftLim h
variable [TopologicalSpace α] [OrderTopology α] | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | leftLim_le_rightLim | null |
tendsto_leftLim (x : α) : Tendsto f (𝓝[<] x) (𝓝 (leftLim f x)) :=
hf.dual_right.tendsto_leftLim x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | tendsto_leftLim | null |
tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≥] leftLim f x) :=
hf.dual_right.tendsto_leftLim_within x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | tendsto_leftLim_within | null |
tendsto_rightLim (x : α) : Tendsto f (𝓝[>] x) (𝓝 (rightLim f x)) :=
hf.dual_right.tendsto_rightLim x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | tendsto_rightLim | null |
tendsto_rightLim_within (x : α) : Tendsto f (𝓝[>] x) (𝓝[≤] rightLim f x) :=
hf.dual_right.tendsto_rightLim_within x | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | tendsto_rightLim_within | null |
continuousWithinAt_Iio_iff_leftLim_eq :
ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x :=
hf.dual_right.continuousWithinAt_Iio_iff_leftLim_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | continuousWithinAt_Iio_iff_leftLim_eq | 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. |
continuousWithinAt_Ioi_iff_rightLim_eq :
ContinuousWithinAt f (Ioi x) x ↔ rightLim f x = f x :=
hf.dual_right.continuousWithinAt_Ioi_iff_rightLim_eq | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | continuousWithinAt_Ioi_iff_rightLim_eq | 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. |
continuousAt_iff_leftLim_eq_rightLim : ContinuousAt f x ↔ leftLim f x = rightLim f x :=
hf.dual_right.continuousAt_iff_leftLim_eq_rightLim | theorem | Topology | [
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.Monotone"
] | Mathlib/Topology/Order/LeftRightLim.lean | continuousAt_iff_leftLim_eq_rightLim | An antitone function is continuous at a point if and only if its left and right limits
coincide. |
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_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩
rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩
exact ⟨u, au, fun x hx => hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, hx.1⟩⟩
tfae_finish | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | TFAE_mem_nhdsGT | 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`. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsGT_iff_exists_mem_Ioc_Ioo_subset | null |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | 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 < u'`, provided `a` is not a top element. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhdsGT_basis_of_exists_gt | null |
nhdsGT_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=
nhdsGT_basis_of_exists_gt <| exists_gt a | lemma | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhdsGT_basis | null |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhdsGT_eq_bot_iff | null |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | 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`. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | countable_setOf_isolated_right | The set of points which are isolated on the right is countable when the space is
second-countable. |
countable_setOf_isolated_left [SecondCountableTopology α] :
{ x : α | 𝓝[<] x = ⊥ }.Countable :=
countable_setOf_isolated_right (α := αᵒᵈ) | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | countable_setOf_isolated_left | The set of points which are isolated on the left is countable when the space is
second-countable. |
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}
suffices H : t.Countable by
have : { x ∈ s | 𝓝[s ∩ Ioi x] x = ⊥ } ⊆ t ∪ {x | IsTop x} := by
intro x hx
by_cases h'x : IsTop x
· simp [h'x]
· simpa [-sep_and, t, h'x]
apply Countable.mono this
simp [H, (subsingleton_isTop α).countable]
have (x) (hx : x ∈ t) : ∃ y > x, s ∩ Ioo x y = ∅ := by
simp only [← empty_mem_iff_bot, mem_nhdsWithin_iff_exists_mem_nhds_inter,
subset_empty_iff, IsTop, not_forall, not_le, mem_setOf_eq, t] at hx
rcases hx.2.1 with ⟨u, hu, h'u⟩
obtain ⟨y, hxy, hy⟩ : ∃ y, x < y ∧ Ico x y ⊆ u := exists_Ico_subset_of_mem_nhds hu hx.2.2
refine ⟨y, hxy, ?_⟩
contrapose! h'u
apply h'u.mono
intro z hz
exact ⟨hy ⟨hz.2.1.le, hz.2.2⟩, hz.1, hz.2.1⟩
choose! y hy h'y using this
apply Set.PairwiseDisjoint.countable_of_Ioo (y := y) _ hy
simp only [PairwiseDisjoint, Set.Pairwise, Function.onFun]
intro a ha b hb hab
wlog H : a < b generalizing a b with h
· have : b < a := lt_of_le_of_ne (not_lt.1 H) hab.symm
exact (h hb ha hab.symm this).symm
have : y a ≤ b := by
by_contra!
have : b ∈ s ∩ Ioo a (y a) := by simp [hb.1, H, this]
simp [h'y a ha] at this
rw [disjoint_iff_forall_ne]
exact fun u hu v hv ↦ ((hu.2.trans_le this).trans hv.1).ne | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | countable_setOf_isolated_right_within | The set of points in a set which are isolated on the right in this set is countable when the
space is second-countable. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | countable_setOf_isolated_left_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. |
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⟩
open List in | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsGT_iff_exists_Ioc_subset | A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. |
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,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`
∃ l ∈ Iio 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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | TFAE_mem_nhdsLT | 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` |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsLT_iff_exists_mem_Ico_Ioo_subset | null |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | 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`, provided `a` is not a bottom element. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | 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`. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsLT_iff_exists_Ico_subset | A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`
with `l < a`. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhdsLT_basis_of_exists_lt | null |
nhdsLT_basis [NoMinOrder α] (a : α) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) :=
nhdsLT_basis_of_exists_lt <| exists_lt a | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhdsLT_basis | null |
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
open List in | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhdsLT_eq_bot_iff | null |
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_le_right _ _⟩,
(Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩
tfae_finish | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | TFAE_mem_nhdsGE | 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`. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsGE_iff_exists_mem_Ioc_Ico_subset | null |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | 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 < u'`, provided `a` is not a top element. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | 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`. |
nhdsGE_basis_Ico [NoMaxOrder α] (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) :=
⟨fun _ => mem_nhdsGE_iff_exists_Ico_subset⟩ | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhdsGE_basis_Ico | null |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhdsGE_basis_Icc | The filter of right neighborhoods has a basis of closed intervals. |
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
open List in | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsGE_iff_exists_Icc_subset | A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
with `a < u`. |
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 : `s` includes `(l, b]` for some `l ∈ [a, b)`
∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b`
simpa using TFAE_mem_nhdsGE h.dual (ofDual ⁻¹' s) | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | TFAE_mem_nhdsLE | 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` |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsLE_iff_exists_mem_Ico_Ioc_subset | null |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | 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`, provided `a` is not a bottom element. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | 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`. |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | mem_nhdsLE_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`. |
nhdsLE_basis_Icc [NoMinOrder α] [DenselyOrdered α] {a : α} :
(𝓝[≤] a).HasBasis (· < a) (Icc · a) :=
⟨fun _ ↦ mem_nhdsLE_iff_exists_Icc_subset⟩ | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhdsLE_basis_Icc | The filter of left neighborhoods has a basis of closed intervals. |
@[to_additive]
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]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhds_eq_iInf_mabs_div | null |
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
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | orderTopology_of_nhds_mabs | null |
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]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | LinearOrderedCommGroup.tendsto_nhds | null |
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | eventually_mabs_div_lt | null |
@[to_additive add_atTop /-- In a linearly ordered additive commutative group with the order
topology, if `f` tends to `C` and `g` tends to `atTop` then `f + g` tends to `atTop`. -/]
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | 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 `atTop` then `f * g` tends to `atTop`. |
@[to_additive add_atBot /-- In a linearly ordered additive commutative group with the order
topology, if `f` tends to `C` and `g` tends to `atBot` then `f + g` tends to `atBot`. -/]
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | Filter.Tendsto.mul_atBot' | 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`. |
@[to_additive atTop_add /-- In a linearly ordered additive commutative group with the order
topology, if `f` tends to `atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/]
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 | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | Filter.Tendsto.atTop_mul' | 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`. |
@[to_additive atBot_add /-- In a linearly ordered additive commutative group with the order
topology, if `f` tends to `atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/]
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
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | Filter.Tendsto.atBot_mul' | 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`. |
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 _ _)⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhds_basis_mabs_div_lt | null |
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]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhds_basis_Ioo_one_lt | null |
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⟩)
variable (α) in
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhds_basis_Icc_one_lt | null |
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)
@[deprecated (since := "2025-03-18")]
alias nhds_basis_zero_abs_sub_lt := nhds_basis_zero_abs_lt | theorem | Topology | [
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Order.Filter.AtTopBot.CompleteLattice",
"Mathlib.Order.Filter.AtTopBot.Group",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/Order/LeftRightNhds.lean | nhds_basis_one_mabs_lt | null |
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