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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AntitoneOn.tendsto_nhdsWithin_Ioo_left {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} {x y : α} (h_nonempty : (Ioo y x).Nonempty) (Af : AntitoneOn f (Ioo y x))
(h_bdd : BddBelow (f '' Ioo y x)) :
... | MonotoneOn.tendsto_nhdsWithin_Ioo_left h_nonempty Af.dual_right h_bdd | lemma | AntitoneOn.tendsto_nhdsWithin_Ioo_left | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"AntitoneOn",
"ConditionallyCompleteLinearOrder",
"LinearOrder",
"MonotoneOn.tendsto_nhdsWithin_Ioo_left",
"OrderTopology",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
AntitoneOn.tendsto_nhdsWithin_Ioo_right {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} {x y : α} (h_nonempty : (Ioo x y).Nonempty) (Af : AntitoneOn f (Ioo x y))
(h_bdd : BddAbove (f '' Ioo x y)) :
... | MonotoneOn.tendsto_nhdsWithin_Ioo_right h_nonempty Af.dual_right h_bdd | lemma | AntitoneOn.tendsto_nhdsWithin_Ioo_right | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"AntitoneOn",
"BddAbove",
"ConditionallyCompleteLinearOrder",
"LinearOrder",
"MonotoneOn.tendsto_nhdsWithin_Ioo_right",
"OrderTopology",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
AntitoneOn.tendsto_nhdsLT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x : α}
(Af : AntitoneOn f (Iio x)) (h_bdd : BddBelow (f '' Iio x)) :
Tendsto f (𝓝[<] x) (𝓝 (sInf (f '' Iio x))) | MonotoneOn.tendsto_nhdsLT Af.dual_right h_bdd | lemma | AntitoneOn.tendsto_nhdsLT | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"AntitoneOn",
"ConditionallyCompleteLinearOrder",
"LinearOrder",
"MonotoneOn.tendsto_nhdsLT",
"OrderTopology",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
AntitoneOn.tendsto_nhdsGT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x : α}
(Af : AntitoneOn f (Ioi x)) (h_bdd : BddAbove (f '' Ioi x)) :
Tendsto f (𝓝[>] x) (𝓝 (sSup (f '' Ioi x))) | MonotoneOn.tendsto_nhdsGT Af.dual_right h_bdd | lemma | AntitoneOn.tendsto_nhdsGT | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"AntitoneOn",
"BddAbove",
"ConditionallyCompleteLinearOrder",
"LinearOrder",
"MonotoneOn.tendsto_nhdsGT",
"OrderTopology",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Antitone.tendsto_nhdsLT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β}
(Af : Antitone f) (x : α) : Tendsto f (𝓝[<] x) (𝓝 (sInf (f '' Iio x))) | Monotone.tendsto_nhdsLT Af.dual_right x | theorem | Antitone.tendsto_nhdsLT | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"ConditionallyCompleteLinearOrder",
"LinearOrder",
"Monotone.tendsto_nhdsLT",
"OrderTopology",
"TopologicalSpace"
] | An antitone map has a limit to the left of any point `x`, equal to `sInf (f '' (Iio x))`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Antitone.tendsto_nhdsGT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β}
(Af : Antitone f) (x : α) : Tendsto f (𝓝[>] x) (𝓝 (sSup (f '' Ioi x))) | Monotone.tendsto_nhdsGT Af.dual_right x | theorem | Antitone.tendsto_nhdsGT | Topology.Order | Mathlib/Topology/Order/Monotone.lean | [] | [
"Antitone",
"ConditionallyCompleteLinearOrder",
"LinearOrder",
"Monotone.tendsto_nhdsGT",
"OrderTopology",
"TopologicalSpace"
] | An antitone map has a limit to the right of any point `x`, equal to `sSup (f '' (Ioi x))`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousWithinAt_right_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) :
ContinuousWithinAt f (Ici a) a | by
have has : a ∈ s := mem_of_mem_nhdsWithin self_mem_Ici hs
refine tendsto_order.2 ⟨fun b hb => ?_, fun b hb => ?_⟩
· filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le
((h_mono.le_iff_le has hxs).2 hxa)
· rcases hfs b hb with ⟨c, hcs, hac, hcb⟩
rw [h_mono.lt_iff... | theorem | StrictMonoOn.continuousWithinAt_right_of_exists_between | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"Set",
"StrictMonoOn",
"mem_of_mem_nhdsWithin",
"self_mem_nhdsWithin"
] | If `f` is a function strictly monotone on a right neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `(f a, b]`, `b > f a`, then `f` is
continuous at `a` from the right.
The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b` is required because otherwise the
function `f : ℝ → ℝ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_right_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
ContinuousWithinAt f (Ici a) a | by
have has : a ∈ s := mem_of_mem_nhdsWithin self_mem_Ici hs
refine tendsto_order.2 ⟨fun b hb => ?_, fun b hb => ?_⟩
· filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le
(h_mono has hxs hxa)
· rcases hfs b hb with ⟨c, hcs, hac, hcb⟩
have : a < c := not_le.1 fun h ... | theorem | continuousWithinAt_right_of_monotoneOn_of_exists_between | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"MonotoneOn",
"Set",
"mem_of_mem_nhdsWithin",
"self_mem_nhdsWithin"
] | If `f` is a monotone function on a right neighborhood of `a` and the image of this neighborhood
under `f` meets every interval `(f a, b)`, `b > f a`, then `f` is continuous at `a` from the right.
The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b` cannot be replaced by the weaker
assumption `hfs : ∀ b > f a, ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a)
(hfs : closure (f '' s) ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a | by
refine continuousWithinAt_right_of_monotoneOn_of_exists_between h_mono hs fun b hb => ?_
rcases (mem_nhdsGE_iff_exists_mem_Ioc_Ico_subset hb).1 hfs with ⟨b', ⟨hab', hbb'⟩, hb'⟩
rcases exists_between hab' with ⟨c', hc'⟩
rcases mem_closure_iff.1 (hb' ⟨hc'.1.le, hc'.2⟩) (Ioo (f a) b') isOpen_Ioo hc' with
⟨_... | theorem | continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"DenselyOrdered",
"MonotoneOn",
"Set",
"closure",
"continuousWithinAt_right_of_monotoneOn_of_exists_between",
"exists_between",
"isOpen_Ioo",
"mem_nhdsGE_iff_exists_mem_Ioc_Ico_subset"
] | If a function `f` with a densely ordered codomain is monotone on a right neighborhood of `a` and
the closure of the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f`
is continuous at `a` from the right. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_right_of_monotoneOn_of_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a)
(hfs : f '' s ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a | continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono hs <|
mem_of_superset hfs subset_closure | theorem | continuousWithinAt_right_of_monotoneOn_of_image_mem_nhdsWithin | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"DenselyOrdered",
"MonotoneOn",
"Set",
"continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin",
"subset_closure"
] | If a function `f` with a densely ordered codomain is monotone on a right neighborhood of `a` and
the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f` is continuous at
`a` from the right. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a)
(hfs : closure (f '' s) ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a | continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin
(fun _ hx _ hy => (h_mono.le_iff_le hx hy).2) hs hfs | theorem | StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"DenselyOrdered",
"Set",
"StrictMonoOn",
"closure",
"continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin"
] | If a function `f` with a densely ordered codomain is strictly monotone on a right neighborhood
of `a` and the closure of the image of this neighborhood under `f` is a right neighborhood of `f a`,
then `f` is continuous at `a` from the right. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousWithinAt_right_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β}
{s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : f '' s ∈ 𝓝[≥] f a) :
ContinuousWithinAt f (Ici a) a | h_mono.continuousWithinAt_right_of_closure_image_mem_nhdsWithin hs
(mem_of_superset hfs subset_closure) | theorem | StrictMonoOn.continuousWithinAt_right_of_image_mem_nhdsWithin | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"DenselyOrdered",
"Set",
"StrictMonoOn",
"subset_closure"
] | If a function `f` with a densely ordered codomain is strictly monotone on a right neighborhood
of `a` and the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f` is
continuous at `a` from the right. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousWithinAt_right_of_surjOn {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : SurjOn f s (Ioi (f a))) :
ContinuousWithinAt f (Ici a) a | h_mono.continuousWithinAt_right_of_exists_between hs fun _ hb =>
let ⟨c, hcs, hcb⟩ := hfs hb
⟨c, hcs, hcb.symm ▸ hb, hcb.le⟩ | theorem | StrictMonoOn.continuousWithinAt_right_of_surjOn | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"Set",
"StrictMonoOn"
] | If a function `f` is strictly monotone on a right neighborhood of `a` and the image of this
neighborhood under `f` includes `Ioi (f a)`, then `f` is continuous at `a` from the right. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousWithinAt_left_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) :
ContinuousWithinAt f (Iic a) a | h_mono.dual.continuousWithinAt_right_of_exists_between hs fun b hb =>
let ⟨c, hcs, hcb, hca⟩ := hfs b hb
⟨c, hcs, hca, hcb⟩ | theorem | StrictMonoOn.continuousWithinAt_left_of_exists_between | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"Set",
"StrictMonoOn"
] | If `f` is a strictly monotone function on a left neighborhood of `a` and the image of this
neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, then `f` is continuous at `a`
from the left.
The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)` is required because otherwise the
function `f : ℝ → ℝ` ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_left_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
(hf : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) :
ContinuousWithinAt f (Iic a) a | @continuousWithinAt_right_of_monotoneOn_of_exists_between αᵒᵈ βᵒᵈ _ _ _ _ _ _ f s a hf.dual hs
fun b hb =>
let ⟨c, hcs, hcb, hca⟩ := hfs b hb
⟨c, hcs, hca, hcb⟩ | theorem | continuousWithinAt_left_of_monotoneOn_of_exists_between | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"MonotoneOn",
"Set",
"continuousWithinAt_right_of_monotoneOn_of_exists_between"
] | If `f` is a monotone function on a left neighborhood of `a` and the image of this neighborhood
under `f` meets every interval `(b, f a)`, `b < f a`, then `f` is continuous at `a` from the left.
The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)` cannot be replaced by the weaker
assumption `hfs : ∀ b < f a, ∃ ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (hf : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a)
(hfs : closure (f '' s) ∈ 𝓝[≤] f a) : ContinuousWithinAt f (Iic a) a | @continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin αᵒᵈ βᵒᵈ _ _ _ _ _ _ _ f s
a hf.dual hs hfs | theorem | continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"DenselyOrdered",
"MonotoneOn",
"Set",
"closure",
"continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin"
] | If a function `f` with a densely ordered codomain is monotone on a left neighborhood of `a` and
the closure of the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is
continuous at `a` from the left | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_left_of_monotoneOn_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β}
{s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] f a) :
ContinuousWithinAt f (Iic a) a | continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono hs
(mem_of_superset hfs subset_closure) | theorem | continuousWithinAt_left_of_monotoneOn_of_image_mem_nhdsWithin | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"DenselyOrdered",
"MonotoneOn",
"Set",
"continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin",
"subset_closure"
] | If a function `f` with a densely ordered codomain is monotone on a left neighborhood of `a` and
the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is continuous at
`a` from the left. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin [DenselyOrdered β]
{f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a)
(hfs : closure (f '' s) ∈ 𝓝[≤] f a) : ContinuousWithinAt f (Iic a) a | h_mono.dual.continuousWithinAt_right_of_closure_image_mem_nhdsWithin hs hfs | theorem | StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"DenselyOrdered",
"Set",
"StrictMonoOn",
"closure"
] | If a function `f` with a densely ordered codomain is strictly monotone on a left neighborhood of
`a` and the closure of the image of this neighborhood under `f` is a left neighborhood of `f a`,
then `f` is continuous at `a` from the left. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousWithinAt_left_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β}
{s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] f a) :
ContinuousWithinAt f (Iic a) a | h_mono.dual.continuousWithinAt_right_of_image_mem_nhdsWithin hs hfs | theorem | StrictMonoOn.continuousWithinAt_left_of_image_mem_nhdsWithin | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"DenselyOrdered",
"Set",
"StrictMonoOn"
] | If a function `f` with a densely ordered codomain is strictly monotone on a left neighborhood of
`a` and the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is
continuous at `a` from the left. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousWithinAt_left_of_surjOn {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : SurjOn f s (Iio (f a))) :
ContinuousWithinAt f (Iic a) a | h_mono.dual.continuousWithinAt_right_of_surjOn hs hfs | theorem | StrictMonoOn.continuousWithinAt_left_of_surjOn | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousWithinAt",
"Set",
"StrictMonoOn"
] | If a function `f` is strictly monotone on a left neighborhood of `a` and the image of this
neighborhood under `f` includes `Iio (f a)`, then `f` is continuous at `a` from the left. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousAt_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a))
(hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) : ContinuousAt f a | continuousAt_iff_continuous_left_right.2
⟨h_mono.continuousWithinAt_left_of_exists_between (mem_nhdsWithin_of_mem_nhds hs) hfs_l,
h_mono.continuousWithinAt_right_of_exists_between (mem_nhdsWithin_of_mem_nhds hs) hfs_r⟩ | theorem | StrictMonoOn.continuousAt_of_exists_between | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousAt",
"Set",
"StrictMonoOn",
"mem_nhdsWithin_of_mem_nhds"
] | If a function `f` is strictly monotone on a neighborhood of `a` and the image of this
neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, and every interval
`(f a, b]`, `b > f a`, then `f` is continuous at `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousAt_of_closure_image_mem_nhds [DenselyOrdered β] {f : α → β}
{s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) : ContinuousAt f a | continuousAt_iff_continuous_left_right.2
⟨h_mono.continuousWithinAt_left_of_closure_image_mem_nhdsWithin (mem_nhdsWithin_of_mem_nhds hs)
(mem_nhdsWithin_of_mem_nhds hfs),
h_mono.continuousWithinAt_right_of_closure_image_mem_nhdsWithin
(mem_nhdsWithin_of_mem_nhds hs) (mem_nhdsWithin_of_mem_nhds... | theorem | StrictMonoOn.continuousAt_of_closure_image_mem_nhds | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousAt",
"DenselyOrdered",
"Set",
"StrictMonoOn",
"closure",
"mem_nhdsWithin_of_mem_nhds"
] | If a function `f` with a densely ordered codomain is strictly monotone on a neighborhood of `a`
and the closure of the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is
continuous at `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
StrictMonoOn.continuousAt_of_image_mem_nhds [DenselyOrdered β] {f : α → β} {s : Set α}
{a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) :
ContinuousAt f a | h_mono.continuousAt_of_closure_image_mem_nhds hs (mem_of_superset hfs subset_closure) | theorem | StrictMonoOn.continuousAt_of_image_mem_nhds | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousAt",
"DenselyOrdered",
"Set",
"StrictMonoOn",
"subset_closure"
] | If a function `f` with a densely ordered codomain is strictly monotone on a neighborhood of `a`
and the image of this set under `f` is a neighborhood of `f a`, then `f` is continuous at `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAt_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a))
(hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) : ContinuousAt f a | continuousAt_iff_continuous_left_right.2
⟨continuousWithinAt_left_of_monotoneOn_of_exists_between h_mono (mem_nhdsWithin_of_mem_nhds hs)
hfs_l,
continuousWithinAt_right_of_monotoneOn_of_exists_between h_mono
(mem_nhdsWithin_of_mem_nhds hs) hfs_r⟩ | theorem | continuousAt_of_monotoneOn_of_exists_between | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousAt",
"MonotoneOn",
"Set",
"continuousWithinAt_right_of_monotoneOn_of_exists_between",
"mem_nhdsWithin_of_mem_nhds"
] | If `f` is a monotone function on a neighborhood of `a` and the image of this neighborhood under
`f` meets every interval `(b, f a)`, `b < f a`, and every interval `(f a, b)`, `b > f a`, then `f`
is continuous at `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAt_of_monotoneOn_of_closure_image_mem_nhds [DenselyOrdered β] {f : α → β}
{s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) : ContinuousAt f a | continuousAt_iff_continuous_left_right.2
⟨continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono
(mem_nhdsWithin_of_mem_nhds hs) (mem_nhdsWithin_of_mem_nhds hfs),
continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono
(mem_nhdsWithin_of_mem_nhds hs)... | theorem | continuousAt_of_monotoneOn_of_closure_image_mem_nhds | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousAt",
"DenselyOrdered",
"MonotoneOn",
"Set",
"closure",
"continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin",
"mem_nhdsWithin_of_mem_nhds"
] | If a function `f` with a densely ordered codomain is monotone on a neighborhood of `a` and the
closure of the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is
continuous at `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAt_of_monotoneOn_of_image_mem_nhds [DenselyOrdered β] {f : α → β} {s : Set α}
{a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) : ContinuousAt f a | continuousAt_of_monotoneOn_of_closure_image_mem_nhds h_mono hs
(mem_of_superset hfs subset_closure) | theorem | continuousAt_of_monotoneOn_of_image_mem_nhds | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"ContinuousAt",
"DenselyOrdered",
"MonotoneOn",
"Set",
"continuousAt_of_monotoneOn_of_closure_image_mem_nhds",
"subset_closure"
] | If a function `f` with a densely ordered codomain is monotone on a neighborhood of `a` and the
image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is continuous at `a`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.continuous_of_denseRange [DenselyOrdered β] {f : α → β} (h_mono : Monotone f)
(h_dense : DenseRange f) : Continuous f | continuous_iff_continuousAt.mpr fun a =>
continuousAt_of_monotoneOn_of_closure_image_mem_nhds (fun _ _ _ _ hxy => h_mono hxy)
univ_mem <|
by simp only [image_univ, h_dense.closure_eq, univ_mem] | theorem | Monotone.continuous_of_denseRange | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"Continuous",
"DenseRange",
"DenselyOrdered",
"Monotone",
"continuousAt_of_monotoneOn_of_closure_image_mem_nhds"
] | A monotone function with densely ordered codomain and a dense range is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Monotone.continuous_of_surjective [DenselyOrdered β] {f : α → β} (h_mono : Monotone f)
(h_surj : Function.Surjective f) : Continuous f | h_mono.continuous_of_denseRange h_surj.denseRange | theorem | Monotone.continuous_of_surjective | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"Continuous",
"DenselyOrdered",
"Monotone"
] | A monotone surjective function with a densely ordered codomain is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuous (e : α ≃o β) : Continuous e | by
rw [‹OrderTopology β›.topology_eq_generate_intervals, continuous_generateFrom_iff]
rintro s ⟨a, rfl | rfl⟩
· rw [e.preimage_Ioi]
apply isOpen_lt'
· rw [e.preimage_Iio]
apply isOpen_gt' | theorem | OrderIso.continuous | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"Continuous",
"continuous_generateFrom_iff",
"isOpen_lt'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toHomeomorph (e : α ≃o β) : α ≃ₜ β | HomeomorphClass.toHomeomorph e | abbrev | OrderIso.toHomeomorph | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [
"HomeomorphClass.toHomeomorph"
] | An order isomorphism between two linear order `OrderTopology` spaces is a homeomorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_toHomeomorph (e : α ≃o β) : ⇑e.toHomeomorph = e | rfl | theorem | OrderIso.coe_toHomeomorph | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_toHomeomorph_symm (e : α ≃o β) : ⇑e.toHomeomorph.symm = e.symm | rfl | theorem | OrderIso.coe_toHomeomorph_symm | Topology.Order | Mathlib/Topology/Order/MonotoneContinuity.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SupConvergenceClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
/-- proof that a monotone function tends to `𝓝 a` as `x → ∞` -/
tendsto_coe_atTop_isLUB :
∀ (a : α) (s : Set α), IsLUB s a → Tendsto ((↑) : s → α) atTop (𝓝 a) | class | SupConvergenceClass | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"IsLUB",
"Preorder",
"Set",
"TopologicalSpace"
] | We say that `α` is a `SupConvergenceClass` if the following holds. Let `f : ι → α` be a
monotone function, let `a : α` be a least upper bound of `Set.range f`. Then `f x` tends to `𝓝 a`
as `x → ∞` (formally, at the filter `Filter.atTop`). We require this for `ι = (s : Set α)`,
`f = (↑)` in the definition, then prove i... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
InfConvergenceClass (α : Type*) [Preorder α] [TopologicalSpace α] : Prop where
/-- proof that a monotone function tends to `𝓝 a` as `x → -∞` -/
tendsto_coe_atBot_isGLB :
∀ (a : α) (s : Set α), IsGLB s a → Tendsto ((↑) : s → α) atBot (𝓝 a) | class | InfConvergenceClass | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Preorder",
"Set",
"TopologicalSpace"
] | We say that `α` is an `InfConvergenceClass` if the following holds. Let `f : ι → α` be a
monotone function, let `a : α` be a greatest lower bound of `Set.range f`. Then `f x` tends to `𝓝 a`
as `x → -∞` (formally, at the filter `Filter.atBot`). We require this for `ι = (s : Set α)`,
`f = (↑)` in the definition, then pr... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
OrderDual.supConvergenceClass [Preorder α] [TopologicalSpace α] [InfConvergenceClass α] :
SupConvergenceClass αᵒᵈ | ⟨‹InfConvergenceClass α›.1⟩ | instance | OrderDual.supConvergenceClass | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"InfConvergenceClass",
"Preorder",
"SupConvergenceClass",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
OrderDual.infConvergenceClass [Preorder α] [TopologicalSpace α] [SupConvergenceClass α] :
InfConvergenceClass αᵒᵈ | ⟨‹SupConvergenceClass α›.1⟩ | instance | OrderDual.infConvergenceClass | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"InfConvergenceClass",
"Preorder",
"SupConvergenceClass",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop_isLUB (h_mono : Monotone f) (ha : IsLUB (Set.range f) a) :
Tendsto f atTop (𝓝 a) | by
suffices Tendsto (rangeFactorization f) atTop atTop from
(SupConvergenceClass.tendsto_coe_atTop_isLUB _ _ ha).comp this
exact h_mono.rangeFactorization.tendsto_atTop_atTop fun b => b.2.imp fun a ha => ha.ge | theorem | tendsto_atTop_isLUB | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"IsLUB",
"Monotone",
"Set.range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atBot_isLUB (h_anti : Antitone f) (ha : IsLUB (Set.range f) a) :
Tendsto f atBot (𝓝 a) | by convert! tendsto_atTop_isLUB h_anti.dual_left ha using 1 | theorem | tendsto_atBot_isLUB | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"IsLUB",
"Set.range",
"tendsto_atTop_isLUB"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atBot_isGLB (h_mono : Monotone f) (ha : IsGLB (Set.range f) a) :
Tendsto f atBot (𝓝 a) | by convert! tendsto_atTop_isLUB h_mono.dual ha.dual using 1 | theorem | tendsto_atBot_isGLB | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Monotone",
"Set.range",
"tendsto_atTop_isLUB"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop_isGLB (h_anti : Antitone f) (ha : IsGLB (Set.range f) a) :
Tendsto f atTop (𝓝 a) | by convert! tendsto_atBot_isLUB h_anti.dual ha.dual using 1 | theorem | tendsto_atTop_isGLB | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"Set.range",
"tendsto_atBot_isLUB"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop_ciSup (h_mono : Monotone f) (hbdd : BddAbove <| range f) :
Tendsto f atTop (𝓝 (⨆ i, f i)) | by
obtain (h | h) := eq_or_ne atTop (⊥ : Filter ι)
· simp [h]
· obtain ⟨h₁, h₂⟩ := Filter.atTop_neBot_iff.mp ⟨h⟩
exact tendsto_atTop_isLUB h_mono <|
h_mono.directed_le.directedOn_range.isLUB_csSup (Set.range_nonempty f) hbdd | theorem | tendsto_atTop_ciSup | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"BddAbove",
"Filter",
"Monotone",
"Set.range_nonempty",
"eq_or_ne",
"tendsto_atTop_isLUB"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atBot_ciSup (h_anti : Antitone f) (hbdd : BddAbove <| range f) :
Tendsto f atBot (𝓝 (⨆ i, f i)) | by convert! tendsto_atTop_ciSup h_anti.dual hbdd.dual using 1 | theorem | tendsto_atBot_ciSup | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"BddAbove",
"tendsto_atTop_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atBot_ciInf (h_mono : Monotone f) (hbdd : BddBelow <| range f) :
Tendsto f atBot (𝓝 (⨅ i, f i)) | by convert! tendsto_atTop_ciSup h_mono.dual hbdd.dual using 1 | theorem | tendsto_atBot_ciInf | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Monotone",
"tendsto_atTop_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop_ciInf (h_anti : Antitone f) (hbdd : BddBelow <| range f) :
Tendsto f atTop (𝓝 (⨅ i, f i)) | by convert! tendsto_atBot_ciSup h_anti.dual hbdd.dual using 1 | theorem | tendsto_atTop_ciInf | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"tendsto_atBot_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop_iSup (h_mono : Monotone f) : Tendsto f atTop (𝓝 (⨆ i, f i)) | tendsto_atTop_ciSup h_mono (OrderTop.bddAbove _) | theorem | tendsto_atTop_iSup | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Monotone",
"OrderTop.bddAbove",
"tendsto_atTop_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atBot_iSup (h_anti : Antitone f) : Tendsto f atBot (𝓝 (⨆ i, f i)) | tendsto_atBot_ciSup h_anti (OrderTop.bddAbove _) | theorem | tendsto_atBot_iSup | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"OrderTop.bddAbove",
"tendsto_atBot_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atBot_iInf (h_mono : Monotone f) : Tendsto f atBot (𝓝 (⨅ i, f i)) | tendsto_atBot_ciInf h_mono (OrderBot.bddBelow _) | theorem | tendsto_atBot_iInf | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Monotone",
"tendsto_atBot_ciInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop_iInf (h_anti : Antitone f) : Tendsto f atTop (𝓝 (⨅ i, f i)) | tendsto_atTop_ciInf h_anti (OrderBot.bddBelow _) | theorem | tendsto_atTop_iInf | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"tendsto_atTop_ciInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Prod.supConvergenceClass
[Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β]
[SupConvergenceClass α] [SupConvergenceClass β] : SupConvergenceClass (α × β) | by
constructor
rintro ⟨a, b⟩ s h
rw [isLUB_prod, ← range_restrict, ← range_restrict] at h
have A : Tendsto (fun x : s => (x : α × β).1) atTop (𝓝 a) :=
tendsto_atTop_isLUB (monotone_fst.restrict s) h.1
have B : Tendsto (fun x : s => (x : α × β).2) atTop (𝓝 b) :=
tendsto_atTop_isLUB (monotone_snd.rest... | instance | Prod.supConvergenceClass | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Preorder",
"SupConvergenceClass",
"TopologicalSpace",
"isLUB_prod",
"tendsto_atTop_isLUB"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.supConvergenceClass
{ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]
[∀ i, SupConvergenceClass (α i)] : SupConvergenceClass (∀ i, α i) | by
refine ⟨fun f s h => ?_⟩
simp only [isLUB_pi, ← range_restrict] at h
exact tendsto_pi_nhds.2 fun i => tendsto_atTop_isLUB ((monotone_eval _).restrict _) (h i) | instance | Pi.supConvergenceClass | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Preorder",
"SupConvergenceClass",
"TopologicalSpace",
"isLUB_pi",
"tendsto_atTop_isLUB"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.infConvergenceClass
{ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)]
[∀ i, InfConvergenceClass (α i)] : InfConvergenceClass (∀ i, α i) | show InfConvergenceClass (∀ i, (α i)ᵒᵈ)ᵒᵈ from OrderDual.infConvergenceClass | instance | Pi.infConvergenceClass | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"InfConvergenceClass",
"OrderDual.infConvergenceClass",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.supConvergenceClass' {ι : Type*} [Preorder α] [TopologicalSpace α]
[SupConvergenceClass α] : SupConvergenceClass (ι → α) | supConvergenceClass | instance | Pi.supConvergenceClass' | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Preorder",
"SupConvergenceClass",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.infConvergenceClass' {ι : Type*} [Preorder α] [TopologicalSpace α]
[InfConvergenceClass α] : InfConvergenceClass (ι → α) | Pi.infConvergenceClass | instance | Pi.infConvergenceClass' | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"InfConvergenceClass",
"Pi.infConvergenceClass",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop_of_monotone {ι α : Type*} [Preorder ι] [TopologicalSpace α]
[ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Monotone f) :
Tendsto f atTop atTop ∨ ∃ l, Tendsto f atTop (𝓝 l) | by
classical
exact if H : BddAbove (range f) then Or.inr ⟨_, tendsto_atTop_ciSup h_mono H⟩
else Or.inl <| tendsto_atTop_atTop_of_monotone' h_mono H
@[deprecated (since := "2026-01-22")] alias tendsto_of_monotone := tendsto_atTop_of_monotone | theorem | tendsto_atTop_of_monotone | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"BddAbove",
"ConditionallyCompleteLinearOrder",
"Monotone",
"OrderTopology",
"Preorder",
"TopologicalSpace",
"tendsto_atTop_ciSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atTop_of_antitone {ι α : Type*} [Preorder ι] [TopologicalSpace α]
[ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Antitone f) :
Tendsto f atTop atBot ∨ ∃ l, Tendsto f atTop (𝓝 l) | tendsto_atTop_of_monotone (α := αᵒᵈ) h_mono
@[deprecated (since := "2026-01-22")] alias tendsto_of_antitone := tendsto_atTop_of_antitone | theorem | tendsto_atTop_of_antitone | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"ConditionallyCompleteLinearOrder",
"OrderTopology",
"Preorder",
"TopologicalSpace",
"tendsto_atTop_of_monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atBot_of_monotone {ι α : Type*} [Preorder ι] [TopologicalSpace α]
[ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Monotone f) :
Tendsto f atBot atBot ∨ ∃ l, Tendsto f atBot (𝓝 l) | tendsto_atTop_of_monotone (ι := ιᵒᵈ) (α := αᵒᵈ) h_mono.dual | theorem | tendsto_atBot_of_monotone | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"ConditionallyCompleteLinearOrder",
"Monotone",
"OrderTopology",
"Preorder",
"TopologicalSpace",
"tendsto_atTop_of_monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_atBot_of_antitone {ι α : Type*} [Preorder ι] [TopologicalSpace α]
[ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Antitone f) :
Tendsto f atBot atTop ∨ ∃ l, Tendsto f atBot (𝓝 l) | tendsto_atTop_of_antitone (ι := ιᵒᵈ) (α := αᵒᵈ) h_mono.dual | theorem | tendsto_atBot_of_antitone | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"ConditionallyCompleteLinearOrder",
"OrderTopology",
"Preorder",
"TopologicalSpace",
"tendsto_atTop_of_antitone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_iff_tendsto_subseq_of_monotone {ι₁ ι₂ α : Type*} [SemilatticeSup ι₁] [Preorder ι₂]
[Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
[NoMaxOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Monotone f)
(hg : Tendsto φ atTop atTop) : Tendsto f atTop (𝓝 l) ↔ Ten... | by
constructor <;> intro h
· exact h.comp hg
· rcases tendsto_atTop_of_monotone hf with (h' | ⟨l', hl'⟩)
· exact (not_tendsto_atTop_of_tendsto_nhds h (h'.comp hg)).elim
· rwa [tendsto_nhds_unique h (hl'.comp hg)] | theorem | tendsto_iff_tendsto_subseq_of_monotone | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"ConditionallyCompleteLinearOrder",
"Monotone",
"NoMaxOrder",
"OrderTopology",
"Preorder",
"SemilatticeSup",
"TopologicalSpace",
"tendsto_atTop_of_monotone",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_iff_tendsto_subseq_of_antitone {ι₁ ι₂ α : Type*} [SemilatticeSup ι₁] [Preorder ι₂]
[Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
[NoMinOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Antitone f)
(hg : Tendsto φ atTop atTop) : Tendsto f atTop (𝓝 l) ↔ Ten... | tendsto_iff_tendsto_subseq_of_monotone (α := αᵒᵈ) hf hg | theorem | tendsto_iff_tendsto_subseq_of_antitone | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"ConditionallyCompleteLinearOrder",
"NoMinOrder",
"OrderTopology",
"Preorder",
"SemilatticeSup",
"TopologicalSpace",
"tendsto_iff_tendsto_subseq_of_monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Monotone.ge_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Preorder β] [IsDirectedOrder β] {f : β → α} {a : α} (hf : Monotone f)
(ha : Tendsto f atTop (𝓝 a)) (b : β) :
f b ≤ a | haveI : Nonempty β := Nonempty.intro b
_root_.ge_of_tendsto ha ((eventually_ge_atTop b).mono fun _ hxy => hf hxy) | theorem | Monotone.ge_of_tendsto | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"IsDirectedOrder",
"Monotone",
"OrderClosedTopology",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Monotone.le_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Preorder β] [IsCodirectedOrder β] {f : β → α} {a : α} (hf : Monotone f)
(ha : Tendsto f atBot (𝓝 a)) (b : β) :
a ≤ f b | hf.dual.ge_of_tendsto ha b | theorem | Monotone.le_of_tendsto | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Monotone",
"OrderClosedTopology",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Antitone.le_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Preorder β] [IsDirectedOrder β] {f : β → α} {a : α} (hf : Antitone f)
(ha : Tendsto f atTop (𝓝 a)) (b : β) :
a ≤ f b | hf.dual_right.ge_of_tendsto ha b | theorem | Antitone.le_of_tendsto | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"IsDirectedOrder",
"OrderClosedTopology",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Antitone.ge_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Preorder β] [IsCodirectedOrder β] {f : β → α} {a : α} (hf : Antitone f)
(ha : Tendsto f atBot (𝓝 a)) (b : β) :
f b ≤ a | hf.dual_right.le_of_tendsto ha b | theorem | Antitone.ge_of_tendsto | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"OrderClosedTopology",
"Preorder",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLUB_of_tendsto_atTop [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Preorder β] [IsDirectedOrder β] [Nonempty β] {f : β → α} {a : α} (hf : Monotone f)
(ha : Tendsto f atTop (𝓝 a)) : IsLUB (Set.range f) a | by
constructor
· rintro _ ⟨b, rfl⟩
exact hf.ge_of_tendsto ha b
· exact fun _ hb => le_of_tendsto' ha fun x => hb (Set.mem_range_self x) | theorem | isLUB_of_tendsto_atTop | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"IsDirectedOrder",
"IsLUB",
"Monotone",
"OrderClosedTopology",
"Preorder",
"Set.mem_range_self",
"Set.range",
"TopologicalSpace",
"le_of_tendsto'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isGLB_of_tendsto_atBot [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Preorder β] [IsCodirectedOrder β] [Nonempty β] {f : β → α} {a : α} (hf : Monotone f)
(ha : Tendsto f atBot (𝓝 a)) : IsGLB (Set.range f) a | isLUB_of_tendsto_atTop (α := αᵒᵈ) (β := βᵒᵈ) hf.dual ha | theorem | isGLB_of_tendsto_atBot | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Monotone",
"OrderClosedTopology",
"Preorder",
"Set.range",
"TopologicalSpace",
"isLUB_of_tendsto_atTop"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLUB_of_tendsto_atBot [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Preorder β] [IsCodirectedOrder β] [Nonempty β] {f : β → α} {a : α} (hf : Antitone f)
(ha : Tendsto f atBot (𝓝 a)) : IsLUB (Set.range f) a | isLUB_of_tendsto_atTop (α := α) (β := βᵒᵈ) hf.dual_left ha | theorem | isLUB_of_tendsto_atBot | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"IsLUB",
"OrderClosedTopology",
"Preorder",
"Set.range",
"TopologicalSpace",
"isLUB_of_tendsto_atTop"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isGLB_of_tendsto_atTop [TopologicalSpace α] [Preorder α] [OrderClosedTopology α]
[Preorder β] [IsDirectedOrder β] [Nonempty β] {f : β → α} {a : α} (hf : Antitone f)
(ha : Tendsto f atTop (𝓝 a)) : IsGLB (Set.range f) a | isGLB_of_tendsto_atBot (α := α) (β := βᵒᵈ) hf.dual_left ha | theorem | isGLB_of_tendsto_atTop | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"IsDirectedOrder",
"OrderClosedTopology",
"Preorder",
"Set.range",
"TopologicalSpace",
"isGLB_of_tendsto_atBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iSup_eq_of_tendsto {α β} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α]
[Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) :
Tendsto f atTop (𝓝 a) → iSup f = a | tendsto_nhds_unique (tendsto_atTop_iSup hf) | theorem | iSup_eq_of_tendsto | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"CompleteLinearOrder",
"Monotone",
"OrderTopology",
"SemilatticeSup",
"TopologicalSpace",
"iSup",
"tendsto_atTop_iSup",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iInf_eq_of_tendsto {α} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α]
[Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) :
Tendsto f atTop (𝓝 a) → iInf f = a | tendsto_nhds_unique (tendsto_atTop_iInf hf) | theorem | iInf_eq_of_tendsto | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"CompleteLinearOrder",
"OrderTopology",
"SemilatticeSup",
"TopologicalSpace",
"tendsto_atTop_iInf",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iSup_eq_iSup_subseq_of_monotone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
{l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f)
(hφ : Tendsto φ l atTop) : ⨆ i, f i = ⨆ i, f (φ i) | le_antisymm
(iSup_mono' fun i =>
Exists.imp (fun j (hj : i ≤ φ j) => hf hj) (hφ.eventually <| eventually_ge_atTop i).exists)
(iSup_mono' fun i => ⟨φ i, le_rfl⟩) | theorem | iSup_eq_iSup_subseq_of_monotone | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"CompleteLattice",
"Filter",
"Monotone",
"Preorder",
"iSup_mono'",
"le_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iSup_eq_iSup_subseq_of_antitone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
{l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f)
(hφ : Tendsto φ l atBot) : ⨆ i, f i = ⨆ i, f (φ i) | le_antisymm
(iSup_mono' fun i =>
Exists.imp (fun j (hj : φ j ≤ i) => hf hj) (hφ.eventually <| eventually_le_atBot i).exists)
(iSup_mono' fun i => ⟨φ i, le_rfl⟩) | theorem | iSup_eq_iSup_subseq_of_antitone | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"CompleteLattice",
"Filter",
"Preorder",
"iSup_mono'",
"le_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iInf_eq_iInf_subseq_of_monotone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
{l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f)
(hφ : Tendsto φ l atBot) : ⨅ i, f i = ⨅ i, f (φ i) | iSup_eq_iSup_subseq_of_monotone hf.dual hφ | theorem | iInf_eq_iInf_subseq_of_monotone | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"CompleteLattice",
"Filter",
"Monotone",
"Preorder",
"iSup_eq_iSup_subseq_of_monotone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iInf_eq_iInf_subseq_of_antitone {ι₁ ι₂ α : Type*} [Preorder ι₂] [CompleteLattice α]
{l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f)
(hφ : Tendsto φ l atTop) : ⨅ i, f i = ⨅ i, f (φ i) | iSup_eq_iSup_subseq_of_antitone hf.dual hφ | theorem | iInf_eq_iInf_subseq_of_antitone | Topology.Order | Mathlib/Topology/Order/MonotoneConvergence.lean | [] | [
"Antitone",
"CompleteLattice",
"Filter",
"Preorder",
"iSup_eq_iSup_subseq_of_antitone"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_Ioi : 𝓝ˢ (Ioi a) = 𝓟 (Ioi a) | isOpen_Ioi.nhdsSet_eq | theorem | nhdsSet_Ioi | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_Iio : 𝓝ˢ (Iio a) = 𝓟 (Iio a) | isOpen_Iio.nhdsSet_eq | theorem | nhdsSet_Iio | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_Ioo : 𝓝ˢ (Ioo a b) = 𝓟 (Ioo a b) | isOpen_Ioo.nhdsSet_eq | theorem | nhdsSet_Ioo | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_Ici : 𝓝ˢ (Ici a) = 𝓝 a ⊔ 𝓟 (Ioi a) | by
rw [← Ioi_insert, nhdsSet_insert, nhdsSet_Ioi] | theorem | nhdsSet_Ici | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"nhdsSet_Ioi",
"nhdsSet_insert"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_Iic : 𝓝ˢ (Iic a) = 𝓝 a ⊔ 𝓟 (Iio a) | nhdsSet_Ici (α := αᵒᵈ) | theorem | nhdsSet_Iic | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"nhdsSet_Ici"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_Ico (h : a < b) : 𝓝ˢ (Ico a b) = 𝓝 a ⊔ 𝓟 (Ioo a b) | by
rw [← Ioo_insert_left h, nhdsSet_insert, nhdsSet_Ioo] | theorem | nhdsSet_Ico | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"nhdsSet_Ioo",
"nhdsSet_insert"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_Ioc (h : a < b) : 𝓝ˢ (Ioc a b) = 𝓝 b ⊔ 𝓟 (Ioo a b) | by
rw [← Ioo_insert_right h, nhdsSet_insert, nhdsSet_Ioo] | theorem | nhdsSet_Ioc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"nhdsSet_Ioo",
"nhdsSet_insert"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsSet_Icc (h : a ≤ b) : 𝓝ˢ (Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Ioo a b) | by
rcases h.eq_or_lt with rfl | hlt
· simp
· rw [← Ioc_insert_left h, nhdsSet_insert, nhdsSet_Ioc hlt, sup_assoc] | theorem | nhdsSet_Icc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"nhdsSet_Ioc",
"nhdsSet_insert",
"sup_assoc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioi_mem_nhdsSet_Ici_iff : Ioi a ∈ 𝓝ˢ (Ici b) ↔ a < b | by
rw [isOpen_Ioi.mem_nhdsSet, Ici_subset_Ioi]
alias ⟨_, Ioi_mem_nhdsSet_Ici⟩ := Ioi_mem_nhdsSet_Ici_iff | theorem | Ioi_mem_nhdsSet_Ici_iff | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ici_mem_nhdsSet_Ici (h : a < b) : Ici a ∈ 𝓝ˢ (Ici b) | mem_of_superset (Ioi_mem_nhdsSet_Ici h) Ioi_subset_Ici_self | theorem | Ici_mem_nhdsSet_Ici | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Iio_mem_nhdsSet_Iic_iff : Iio b ∈ 𝓝ˢ (Iic a) ↔ a < b | Ioi_mem_nhdsSet_Ici_iff (α := αᵒᵈ)
alias ⟨_, Iio_mem_nhdsSet_Iic⟩ := Iio_mem_nhdsSet_Iic_iff | theorem | Iio_mem_nhdsSet_Iic_iff | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ioi_mem_nhdsSet_Ici_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Iic_mem_nhdsSet_Iic (h : a < b) : Iic b ∈ 𝓝ˢ (Iic a) | Ici_mem_nhdsSet_Ici (α := αᵒᵈ) h | theorem | Iic_mem_nhdsSet_Iic | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ici_mem_nhdsSet_Ici"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioi_mem_nhdsSet_Icc (h : a < b) : Ioi a ∈ 𝓝ˢ (Icc b c) | nhdsSet_mono Icc_subset_Ici_self <| Ioi_mem_nhdsSet_Ici h | theorem | Ioi_mem_nhdsSet_Icc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"nhdsSet_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ici_mem_nhdsSet_Icc (h : a < b) : Ici a ∈ 𝓝ˢ (Icc b c) | mem_of_superset (Ioi_mem_nhdsSet_Icc h) Ioi_subset_Ici_self | theorem | Ici_mem_nhdsSet_Icc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ioi_mem_nhdsSet_Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Iio_mem_nhdsSet_Icc (h : b < c) : Iio c ∈ 𝓝ˢ (Icc a b) | nhdsSet_mono Icc_subset_Iic_self <| Iio_mem_nhdsSet_Iic h | theorem | Iio_mem_nhdsSet_Icc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"nhdsSet_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Iic_mem_nhdsSet_Icc (h : b < c) : Iic c ∈ 𝓝ˢ (Icc a b) | mem_of_superset (Iio_mem_nhdsSet_Icc h) Iio_subset_Iic_self | theorem | Iic_mem_nhdsSet_Icc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iio_mem_nhdsSet_Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioo_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ioo a d ∈ 𝓝ˢ (Icc b c) | inter_mem (Ioi_mem_nhdsSet_Icc h) (Iio_mem_nhdsSet_Icc h') | theorem | Ioo_mem_nhdsSet_Icc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iio_mem_nhdsSet_Icc",
"Ioi_mem_nhdsSet_Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ico_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ico a d ∈ 𝓝ˢ (Icc b c) | inter_mem (Ici_mem_nhdsSet_Icc h) (Iio_mem_nhdsSet_Icc h') | theorem | Ico_mem_nhdsSet_Icc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ici_mem_nhdsSet_Icc",
"Iio_mem_nhdsSet_Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioc_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ioc a d ∈ 𝓝ˢ (Icc b c) | inter_mem (Ioi_mem_nhdsSet_Icc h) (Iic_mem_nhdsSet_Icc h') | theorem | Ioc_mem_nhdsSet_Icc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iic_mem_nhdsSet_Icc",
"Ioi_mem_nhdsSet_Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Icc_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Icc a d ∈ 𝓝ˢ (Icc b c) | inter_mem (Ici_mem_nhdsSet_Icc h) (Iic_mem_nhdsSet_Icc h') | theorem | Icc_mem_nhdsSet_Icc | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ici_mem_nhdsSet_Icc",
"Iic_mem_nhdsSet_Icc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ici_mem_nhdsSet_Ico (h : a < b) : Ici a ∈ 𝓝ˢ (Ico b c) | nhdsSet_mono Ico_subset_Icc_self <| Ici_mem_nhdsSet_Icc h | theorem | Ici_mem_nhdsSet_Ico | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ici_mem_nhdsSet_Icc",
"nhdsSet_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioi_mem_nhdsSet_Ico (h : a < b) : Ioi a ∈ 𝓝ˢ (Ico b c) | nhdsSet_mono Ico_subset_Icc_self <| Ioi_mem_nhdsSet_Icc h | theorem | Ioi_mem_nhdsSet_Ico | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ioi_mem_nhdsSet_Icc",
"nhdsSet_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Iio_mem_nhdsSet_Ico (h : b ≤ c) : Iio c ∈ 𝓝ˢ (Ico a b) | nhdsSet_mono Ico_subset_Iio_self <| by simpa | theorem | Iio_mem_nhdsSet_Ico | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"nhdsSet_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Iic_mem_nhdsSet_Ico (h : b ≤ c) : Iic c ∈ 𝓝ˢ (Ico a b) | mem_of_superset (Iio_mem_nhdsSet_Ico h) Iio_subset_Iic_self | theorem | Iic_mem_nhdsSet_Ico | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iio_mem_nhdsSet_Ico"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioo_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ioo a d ∈ 𝓝ˢ (Ico b c) | inter_mem (Ioi_mem_nhdsSet_Ico h) (Iio_mem_nhdsSet_Ico h') | theorem | Ioo_mem_nhdsSet_Ico | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iio_mem_nhdsSet_Ico",
"Ioi_mem_nhdsSet_Ico"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Icc_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Icc a d ∈ 𝓝ˢ (Ico b c) | inter_mem (Ici_mem_nhdsSet_Ico h) (Iic_mem_nhdsSet_Ico h') | theorem | Icc_mem_nhdsSet_Ico | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Ici_mem_nhdsSet_Ico",
"Iic_mem_nhdsSet_Ico"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Ioc_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ioc a d ∈ 𝓝ˢ (Ico b c) | inter_mem (Ioi_mem_nhdsSet_Ico h) (Iic_mem_nhdsSet_Ico h') | theorem | Ioc_mem_nhdsSet_Ico | Topology.Order | Mathlib/Topology/Order/NhdsSet.lean | [] | [
"Iic_mem_nhdsSet_Ico",
"Ioi_mem_nhdsSet_Ico"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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