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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isLocallyClosed_Ico [LinearOrder X] [ClosedIciTopology X] :
IsLocallyClosed (Set.Ico a b) | by
rw [← Set.Iio_inter_Ici]
exact isLocallyClosed_Iio.inter isLocallyClosed_Ici | theorem | isLocallyClosed_Ico | Topology.Order | Mathlib/Topology/Order/IsLocallyClosed.lean | [] | [
"ClosedIciTopology",
"IsLocallyClosed",
"LinearOrder",
"Set.Ico",
"isLocallyClosed_Ici"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :
∃ᶠ x in 𝓝[≤] a, x ∈ s | by
rcases hs with ⟨a', ha'⟩
intro h
rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)
· exact h.self_of_nhdsWithin le_rfl ha'
· rcases (mem_nhdsLE_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩
rcases ha.exists_between hba with ⟨b', hb's, hb'⟩
exact hb hb' hb's | theorem | IsLUB.frequently_mem | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB",
"Set",
"le_rfl",
"mem_nhdsLE_iff_exists_Ioc_subset'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :
∃ᶠ x in 𝓝 a, x ∈ s | (ha.frequently_mem hs).filter_mono inf_le_left | theorem | IsLUB.frequently_nhds_mem | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :
∃ᶠ x in 𝓝[≥] a, x ∈ s | IsLUB.frequently_mem (α := αᵒᵈ) ha hs | theorem | IsGLB.frequently_mem | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB.frequently_mem",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :
∃ᶠ x in 𝓝 a, x ∈ s | (ha.frequently_mem hs).filter_mono inf_le_left | theorem | IsGLB.frequently_nhds_mem | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s | (ha.frequently_nhds_mem hs).mem_closure | theorem | IsLUB.mem_closure | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB",
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s | (ha.frequently_nhds_mem hs).mem_closure | theorem | IsGLB.mem_closure | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :
NeBot (𝓝[s] a) | mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs) | theorem | IsLUB.nhdsWithin_neBot | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :
NeBot (𝓝[s] a) | IsLUB.nhdsWithin_neBot (α := αᵒᵈ) ha hs | theorem | IsGLB.nhdsWithin_neBot | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB.nhdsWithin_neBot",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)
[NeBot (f ⊓ 𝓝 a)] : IsLUB s a | ⟨hsa, fun b hb =>
not_lt.1 fun hba =>
have : s ∩ { a | b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba)
let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this
have : b < b := lt_of_lt_of_le hxb <| hb hxs
lt_irrefl b this⟩ | theorem | isLUB_of_mem_nhds | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Filter",
"Filter.nonempty_of_mem",
"IsLUB",
"IsOpen.mem_nhds",
"Set",
"isOpen_lt'",
"lt_irrefl",
"lt_of_lt_of_le",
"upperBounds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :
IsLUB s a | by
rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf
exact isLUB_of_mem_nhds hsa (mem_principal_self s) | theorem | isLUB_of_mem_closure | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"ClusterPt",
"IsLUB",
"Set",
"closure",
"isLUB_of_mem_nhds",
"mem_closure_iff_clusterPt",
"upperBounds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isGLB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ lowerBounds s) (hsf : s ∈ f)
[NeBot (f ⊓ 𝓝 a)] :
IsGLB s a | isLUB_of_mem_nhds (α := αᵒᵈ) hsa hsf | theorem | isGLB_of_mem_nhds | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Filter",
"Set",
"isLUB_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :
IsGLB s a | isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf | theorem | isGLB_of_mem_closure | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Set",
"closure",
"isLUB_of_mem_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
{f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) | by
rintro _ ⟨x, hx, rfl⟩
replace ha := ha.inter_Ici_of_mem hx
haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩
refine ge_of_tendsto (hb.mono_left (nhdsWithin_mono a (inter_subset_left (t := Ici x)))) ?_
exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2 | theorem | IsLUB.mem_upperBounds_of_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB",
"MonotoneOn",
"OrderClosedTopology",
"Preorder",
"Set",
"TopologicalSpace",
"nhdsWithin_mono",
"self_mem_nhdsWithin",
"upperBounds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
{s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b | haveI := ha.nhdsWithin_neBot hs
⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>
le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩ | theorem | IsLUB.isLUB_of_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB",
"MonotoneOn",
"OrderClosedTopology",
"Preorder",
"Set",
"TopologicalSpace",
"le_of_tendsto",
"self_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
{f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) | IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb | theorem | IsGLB.mem_lowerBounds_of_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB.mem_upperBounds_of_tendsto",
"MonotoneOn",
"OrderClosedTopology",
"Preorder",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
{s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) :
IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b | IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual | theorem | IsGLB.isGLB_of_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB.isLUB_of_tendsto",
"MonotoneOn",
"OrderClosedTopology",
"Preorder",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
{f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) | IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb | theorem | IsLUB.mem_lowerBounds_of_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"AntitoneOn",
"IsLUB",
"IsLUB.mem_upperBounds_of_tendsto",
"OrderClosedTopology",
"Preorder",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
{s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsGLB (f '' s) b | IsLUB.isLUB_of_tendsto (γ := γᵒᵈ) hf ha hs hb | theorem | IsLUB.isGLB_of_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"AntitoneOn",
"IsLUB",
"IsLUB.isLUB_of_tendsto",
"OrderClosedTopology",
"Preorder",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
{f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) | IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb | theorem | IsGLB.mem_upperBounds_of_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"AntitoneOn",
"IsGLB.mem_lowerBounds_of_tendsto",
"OrderClosedTopology",
"Preorder",
"Set",
"TopologicalSpace",
"upperBounds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
{s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a) (hs : s.Nonempty)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b | IsGLB.isGLB_of_tendsto (γ := γᵒᵈ) hf ha hs hb | theorem | IsGLB.isLUB_of_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"AntitoneOn",
"IsGLB.isGLB_of_tendsto",
"IsLUB",
"OrderClosedTopology",
"Preorder",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)
(sc : IsClosed s) : a ∈ s | sc.closure_subset <| ha.mem_closure hs
alias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed | theorem | IsLUB.mem_of_isClosed | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsClosed",
"IsLUB",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty)
(sc : IsClosed s) : a ∈ s | sc.closure_subset <| ha.mem_closure hs
alias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed | theorem | IsGLB.mem_of_isClosed | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isLUB_iff_of_subset_of_subset_closure {α : Type*} [TopologicalSpace α] [Preorder α]
[ClosedIicTopology α] {s t : Set α} (hst : s ⊆ t) (hts : t ⊆ closure s) {x : α} :
IsLUB s x ↔ IsLUB t x | isLUB_congr <| (upperBounds_closure (s := s) ▸ upperBounds_mono_set hts).antisymm <|
upperBounds_mono_set hst | theorem | isLUB_iff_of_subset_of_subset_closure | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"ClosedIicTopology",
"IsLUB",
"Preorder",
"Set",
"TopologicalSpace",
"antisymm",
"closure",
"isLUB_congr",
"upperBounds_closure",
"upperBounds_mono_set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isGLB_iff_of_subset_of_subset_closure {α : Type*} [TopologicalSpace α] [Preorder α]
[ClosedIciTopology α] {s t : Set α} (hst : s ⊆ t) (hts : t ⊆ closure s) {x : α} :
IsGLB s x ↔ IsGLB t x | isLUB_iff_of_subset_of_subset_closure (α := αᵒᵈ) hst hts | theorem | isGLB_iff_of_subset_of_subset_closure | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"ClosedIciTopology",
"Preorder",
"Set",
"TopologicalSpace",
"closure",
"isLUB_iff_of_subset_of_subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.isLUB_inter_iff {α : Type*} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α]
{s t : Set α} (hs : Dense s) (ht : IsOpen t) {x : α} :
IsLUB (t ∩ s) x ↔ IsLUB t x | isLUB_iff_of_subset_of_subset_closure (by simp) <| hs.open_subset_closure_inter ht | theorem | Dense.isLUB_inter_iff | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"ClosedIicTopology",
"Dense",
"IsLUB",
"IsOpen",
"Preorder",
"Set",
"TopologicalSpace",
"isLUB_iff_of_subset_of_subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.isGLB_inter_iff {α : Type*} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α]
{s t : Set α} (hs : Dense s) (ht : IsOpen t) {x : α} :
IsGLB (t ∩ s) x ↔ IsGLB t x | hs.isLUB_inter_iff (α := αᵒᵈ) ht | theorem | Dense.isGLB_inter_iff | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"ClosedIciTopology",
"Dense",
"IsOpen",
"Preorder",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.upperBounds_image {α : Type*} [TopologicalSpace α] [Preorder α]
[ClosedIicTopology α] {f : γ → α} [TopologicalSpace γ] {S : Set γ} (hS : Dense S)
(hf : Continuous f) :
upperBounds (f '' S) = upperBounds (range f) | by
refine subset_antisymm ?_ fun _ => upperBounds_mono (Set.image_subset_range f S) le_rfl
refine subset_trans ?_ fun _ => upperBounds_mono (hf.range_subset_closure_image_dense hS) le_rfl
intro x hx i hi
rw [mem_closure_iff_frequently] at hi
exact (hi.mono hx).mem_of_closed isClosed_Iic | theorem | Dense.upperBounds_image | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"ClosedIicTopology",
"Continuous",
"Dense",
"Preorder",
"Set",
"Set.image_subset_range",
"TopologicalSpace",
"isClosed_Iic",
"le_rfl",
"mem_closure_iff_frequently",
"subset_antisymm",
"subset_trans",
"upperBounds",
"upperBounds_mono"
] | The upper bounds of the image of a continuous function on a dense set are equal to the upper
bounds of the range of the universe. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Dense.lowerBounds_image {α : Type*} [TopologicalSpace α] [Preorder α]
[ClosedIciTopology α] {f : γ → α} [TopologicalSpace γ] {S : Set γ} (hS : Dense S)
(hf : Continuous f) :
lowerBounds (f '' S) = lowerBounds (range f) | hS.upperBounds_image (α := αᵒᵈ) hf | theorem | Dense.lowerBounds_image | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"ClosedIciTopology",
"Continuous",
"Dense",
"Preorder",
"Set",
"TopologicalSpace"
] | The lower bounds of the image of a continuous function on a dense set are equal to the lower
bounds of the range of the universe. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Dense.ciSup {α : Type*} [TopologicalSpace α]
[ConditionallyCompleteLattice α] [ClosedIicTopology α] {f : γ → α} [TopologicalSpace γ]
{S : Set γ} (hS : Dense S) (hf : Continuous f) (h : BddAbove (range f)) :
⨆ s : S, f s = ⨆ i, f i | by
rw [← sSup_range, ← sSup_range]
obtain (_ | _) := isEmpty_or_nonempty γ
· simp [Set.range_eq_empty]
refine ((isLUB_csSup (range_nonempty f) h).unique ?_).symm
refine (isLUB_congr (hS.upperBounds_image hf)).mp (isLUB_ciSup_set ?_ hS.nonempty)
exact h.mono (by grind) | theorem | Dense.ciSup | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"BddAbove",
"ClosedIicTopology",
"ConditionallyCompleteLattice",
"Continuous",
"Dense",
"Set",
"Set.range_eq_empty",
"TopologicalSpace",
"isEmpty_or_nonempty",
"isLUB_ciSup_set",
"isLUB_congr",
"isLUB_csSup",
"sSup_range",
"symm"
] | The supremum of a bounded above, continuous function on a dense set is equal to the supremum on
the universe. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Dense.ciInf {α : Type*} [TopologicalSpace α]
[ConditionallyCompleteLattice α] [ClosedIciTopology α] {f : γ → α} [TopologicalSpace γ]
{S : Set γ} (hS : Dense S) (hf : Continuous f) (h : BddBelow (range f)) :
⨅ s : S, f s = ⨅ i, f i | hS.ciSup (α := αᵒᵈ) hf h | theorem | Dense.ciInf | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"ClosedIciTopology",
"ConditionallyCompleteLattice",
"Continuous",
"Dense",
"Set",
"TopologicalSpace"
] | The infimum of a bounded below, continuous function on a dense set is equal to the infimum on
the universe. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Dense.ciSup' {α : Type*} [TopologicalSpace α]
[ConditionallyCompleteLinearOrder α] [ClosedIicTopology α] {f : γ → α} [TopologicalSpace γ]
{S : Set γ} (hS : Dense S) (hf : Continuous f) :
⨆ s : S, f s = ⨆ i, f i | by
by_cases h : BddAbove (range (fun x : S ↦ f x))
· refine hS.ciSup hf <| h.closure.mono ?_
simpa [← Function.comp_def, range_comp] using hf.range_subset_closure_image_dense hS
· suffices ¬ BddAbove (range f) by simp [ciSup_of_not_bddAbove, this, h]
contrapose h
grind [h.mono] | theorem | Dense.ciSup' | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"BddAbove",
"ClosedIicTopology",
"ConditionallyCompleteLinearOrder",
"Continuous",
"Dense",
"Set",
"TopologicalSpace",
"by_cases",
"ciSup_of_not_bddAbove"
] | This is an analogue of `Dense.continuous_sup` for functions taking values in a conditionally
complete linear order. The assumption of `BddAbove (range f)` is not needed in this theorem. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Dense.ciInf' {α : Type*} [TopologicalSpace α]
[ConditionallyCompleteLinearOrder α] [ClosedIciTopology α] {f : γ → α} [TopologicalSpace γ]
{S : Set γ} (hS : Dense S) (hf : Continuous f) :
⨅ s : S, f s = ⨅ i, f i | hS.ciSup' (α := αᵒᵈ) hf | theorem | Dense.ciInf' | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"ClosedIciTopology",
"ConditionallyCompleteLinearOrder",
"Continuous",
"Dense",
"Set",
"TopologicalSpace"
] | This is an analogue of `Dense.continuous_inf` for functions taking values in a conditionally
complete linear order. The assumption of `BddBelow (range f)` is not needed in this theorem. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ConditionallyCompleteLinearOrder.isCompact_Icc (a b : α) :
IsCompact (Icc a b) | by
simp only [isCompact_iff_ultrafilter_le_nhds, le_principal_iff]
refine (le_or_gt a b).elim (fun _ f hfab ↦ ?_) (by simp [·])
by_contra! hf
have hpt : ∀ x ∈ Icc a b, {x} ∉ f := fun x hx _ ↦ hf x hx (le_trans (by simpa) (pure_le_nhds x))
set s := { x ∈ Icc a b | Icc a x ∉ f }
have hsb : b ∈ upperBounds s :... | lemma | ConditionallyCompleteLinearOrder.isCompact_Icc | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsCompact",
"IsLUB",
"isCompact_iff_ultrafilter_le_nhds",
"isLUB_csSup",
"le_of_not_gt",
"le_or_gt",
"le_trans",
"nhds_eq_order",
"notMem_of_csSup_lt",
"pure_le_nhds",
"upperBounds"
] | A closed interval in a conditionally complete linear order is compact.
Also see general API on `CompactIccSpace`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
upperClosure_eq_Ici_csInf {s : Set α} (h₁ : s.Nonempty) (h₂ : BddBelow s) (hs : IsClosed s) :
upperClosure s = Ici (sInf s) | Set.ext fun _ ↦ ⟨fun ⟨_, h, h'⟩ ↦ csInf_le_of_le h₂ h h',
(⟨_, (isGLB_csInf h₁ h₂).mem_of_isClosed h₁ hs, ·⟩)⟩ | lemma | upperClosure_eq_Ici_csInf | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsClosed",
"Set",
"Set.ext",
"upperClosure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lowerClosure_eq_Iic_csSup {s : Set α} (h₁ : s.Nonempty) (h₂ : BddAbove s) (hs : IsClosed s) :
lowerClosure s = Iic (sSup s) | upperClosure_eq_Ici_csInf (α := αᵒᵈ) h₁ h₂ hs | lemma | lowerClosure_eq_Iic_csSup | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"BddAbove",
"IsClosed",
"Set",
"upperClosure_eq_Ici_csInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.upperClosure {s : Set α} (hs : IsClosed s) :
IsClosed (upperClosure s : Set α) | by
obtain rfl | h₁ := s.eq_empty_or_nonempty
· simp
by_cases h₂ : BddBelow s
· exact upperClosure_eq_Ici_csInf h₁ h₂ hs ▸ isClosed_Ici
· exact upperClosure_eq_bot h₂ ▸ isClosed_univ | lemma | IsClosed.upperClosure | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsClosed",
"Set",
"by_cases",
"isClosed_univ",
"upperClosure",
"upperClosure_eq_Ici_csInf",
"upperClosure_eq_bot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.lowerClosure {s : Set α} (hs : IsClosed s) :
IsClosed (lowerClosure s).1 | IsClosed.upperClosure (α := αᵒᵈ) hs | lemma | IsClosed.lowerClosure | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsClosed",
"IsClosed.upperClosure",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.exists_seq_strictMono_tendsto_of_notMem {t : Set α} {x : α}
[IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (notMem : x ∉ t) (ht : t.Nonempty) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t | by
obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht)
replace hvx := hvx.mono_right nhdsWithin_le_nhds
have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) notMem)
have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx')
ch... | theorem | IsLUB.exists_seq_strictMono_tendsto_of_notMem | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB",
"Set",
"StrictMono",
"lt_mem_nhds",
"nhdsWithin_le_nhds",
"strictMono_nat_of_lt_succ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]
(htx : IsLUB t x) (ht : t.Nonempty) :
∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t | by
by_cases h : x ∈ t
· exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩
· rcases htx.exists_seq_strictMono_tendsto_of_notMem h ht with ⟨u, hu⟩
exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩ | theorem | IsLUB.exists_seq_monotone_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB",
"Monotone",
"Set",
"by_cases",
"le_rfl",
"monotone_const",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_strictMono_tendsto' {α : Type*} [LinearOrder α] [TopologicalSpace α]
[DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) | by
have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim
have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy
rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_notMem hx ht with ⟨u, hu⟩
exact ⟨u, hu.1, hu.2.2.symm⟩ | theorem | exists_seq_strictMono_tendsto' | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"DenselyOrdered",
"FirstCountableTopology",
"LinearOrder",
"OrderTopology",
"Set.Nonempty",
"StrictMono",
"TopologicalSpace",
"isLUB_Ioo",
"lt_irrefl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α]
(x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) | by
obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x
rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩
exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩ | theorem | exists_seq_strictMono_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"DenselyOrdered",
"FirstCountableTopology",
"NoMinOrder",
"StrictMono",
"exists_seq_strictMono_tendsto'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α]
[FirstCountableTopology α] (x : α) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) | let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x
⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <| tendsto_nhdsWithin_range.2 h⟩ | theorem | exists_seq_strictMono_tendsto_nhdsWithin | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"DenselyOrdered",
"FirstCountableTopology",
"NoMinOrder",
"StrictMono",
"exists_seq_strictMono_tendsto",
"tendsto_nhdsWithin_mono_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_tendsto_sSup {α : Type*} [ConditionallyCompleteLinearOrder α]
[TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)
(hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S | by
rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩
exact ⟨u, hu.1, hu.2.2⟩ | theorem | exists_seq_tendsto_sSup | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"BddAbove",
"ConditionallyCompleteLinearOrder",
"FirstCountableTopology",
"Monotone",
"OrderTopology",
"Set",
"TopologicalSpace",
"isLUB_csSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.exists_seq_strictMono_tendsto_of_lt [DenselyOrdered α] [FirstCountableTopology α]
{s : Set α} (hs : Dense s) {x y : α} (hy : y < x) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ (Ioo y x ∩ s)) ∧ Tendsto u atTop (𝓝 x) | by
have hnonempty : (Ioo y x ∩ s).Nonempty := by
obtain ⟨z, hyz, hzx⟩ := hs.exists_between hy
exact ⟨z, mem_inter hzx hyz⟩
have hx : IsLUB (Ioo y x ∩ s) x := hs.isLUB_inter_iff isOpen_Ioo |>.mpr <| isLUB_Ioo hy
apply hx.exists_seq_strictMono_tendsto_of_notMem (by simp) hnonempty |>.imp
simp_all | theorem | Dense.exists_seq_strictMono_tendsto_of_lt | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Dense",
"DenselyOrdered",
"FirstCountableTopology",
"IsLUB",
"Set",
"StrictMono",
"isLUB_Ioo",
"isOpen_Ioo"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α]
[FirstCountableTopology α] {s : Set α} (hs : Dense s) (x : α) :
∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ (Iio x ∩ s)) ∧ Tendsto u atTop (𝓝 x) | by
obtain ⟨y, hy⟩ := exists_lt x
apply hs.exists_seq_strictMono_tendsto_of_lt (exists_lt x).choose_spec |>.imp
simp_all | theorem | Dense.exists_seq_strictMono_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Dense",
"DenselyOrdered",
"FirstCountableTopology",
"NoMinOrder",
"Set",
"StrictMono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
DenseRange.exists_seq_strictMono_tendsto_of_lt {β : Type*} [LinearOrder β]
[DenselyOrdered α] [FirstCountableTopology α] {f : β → α} {x y : α} (hf : DenseRange f)
(hmono : Monotone f) (hlt : y < x) :
∃ u : ℕ → β, StrictMono u ∧ (∀ n, f (u n) ∈ Ioo y x) ∧ Tendsto (f ∘ u) atTop (𝓝 x) | by
rcases Dense.exists_seq_strictMono_tendsto_of_lt hf hlt with ⟨u, hu, huyxf, hlim⟩
have huyx (n : ℕ) : u n ∈ Ioo y x := (huyxf n).1
have huf (n : ℕ) : u n ∈ range f := (huyxf n).2
choose v hv using huf
obtain rfl : f ∘ v = u := funext hv
exact ⟨v, fun a b hlt ↦ hmono.reflect_lt <| hu hlt, huyx, hlim⟩ | theorem | DenseRange.exists_seq_strictMono_tendsto_of_lt | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Dense.exists_seq_strictMono_tendsto_of_lt",
"DenseRange",
"DenselyOrdered",
"FirstCountableTopology",
"LinearOrder",
"Monotone",
"StrictMono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
DenseRange.exists_seq_strictMono_tendsto {β : Type*} [LinearOrder β] [DenselyOrdered α]
[NoMinOrder α] [FirstCountableTopology α] {f : β → α} (hf : DenseRange f) (hmono : Monotone f)
(x : α) :
∃ u : ℕ → β, StrictMono u ∧ (∀ n, f (u n) ∈ Iio x) ∧ Tendsto (f ∘ u) atTop (𝓝 x) | by
rcases Dense.exists_seq_strictMono_tendsto hf x with ⟨u, hu, huxf, hlim⟩
have hux (n : ℕ) : u n ∈ Iio x := (huxf n).1
have huf (n : ℕ) : u n ∈ range f := (huxf n).2
choose v hv using huf
obtain rfl : f ∘ v = u := funext hv
exact ⟨v, fun a b hlt ↦ hmono.reflect_lt <| hu hlt, hux, hlim⟩ | theorem | DenseRange.exists_seq_strictMono_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Dense.exists_seq_strictMono_tendsto",
"DenseRange",
"DenselyOrdered",
"FirstCountableTopology",
"LinearOrder",
"Monotone",
"NoMinOrder",
"StrictMono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.exists_seq_strictAnti_tendsto_of_notMem {t : Set α} {x : α}
[IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (notMem : x ∉ t) (ht : t.Nonempty) :
∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t | IsLUB.exists_seq_strictMono_tendsto_of_notMem (α := αᵒᵈ) htx notMem ht | theorem | IsGLB.exists_seq_strictAnti_tendsto_of_notMem | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"IsLUB.exists_seq_strictMono_tendsto_of_notMem",
"Set",
"StrictAnti"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]
(htx : IsGLB t x) (ht : t.Nonempty) :
∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t | IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht | theorem | IsGLB.exists_seq_antitone_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Antitone",
"IsLUB.exists_seq_monotone_tendsto",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α}
(hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) | by
simpa using! exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy) | theorem | exists_seq_strictAnti_tendsto' | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"DenselyOrdered",
"FirstCountableTopology",
"StrictAnti",
"exists_seq_strictMono_tendsto'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α]
(x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) | exists_seq_strictMono_tendsto (α := αᵒᵈ) x | theorem | exists_seq_strictAnti_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"DenselyOrdered",
"FirstCountableTopology",
"NoMaxOrder",
"StrictAnti",
"exists_seq_strictMono_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α]
[FirstCountableTopology α] (x : α) :
∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) | exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _ | theorem | exists_seq_strictAnti_tendsto_nhdsWithin | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"DenselyOrdered",
"FirstCountableTopology",
"NoMaxOrder",
"StrictAnti",
"exists_seq_strictMono_tendsto_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α]
{x y : α} (h : x < y) :
∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧
(∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) | by
rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩
rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩
exact
⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,
fun k l => (hu_anti.antitone zero_le).trans_lt (hv_... | theorem | exists_seq_strictAnti_strictMono_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"DenselyOrdered",
"FirstCountableTopology",
"StrictAnti",
"StrictMono",
"exists_seq_strictAnti_tendsto'",
"exists_seq_strictMono_tendsto'",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_seq_tendsto_sInf {α : Type*} [ConditionallyCompleteLinearOrder α]
[TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty)
(hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S | exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS' | theorem | exists_seq_tendsto_sInf | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Antitone",
"ConditionallyCompleteLinearOrder",
"FirstCountableTopology",
"OrderTopology",
"Set",
"TopologicalSpace",
"exists_seq_tendsto_sSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.exists_seq_strictAnti_tendsto_of_lt [DenselyOrdered α] [FirstCountableTopology α]
{s : Set α} (hs : Dense s) {x y : α} (hy : x < y) :
∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ (Ioo x y ∩ s)) ∧ Tendsto u atTop (𝓝 x) | by
simpa using! hs.exists_seq_strictMono_tendsto_of_lt (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy) | theorem | Dense.exists_seq_strictAnti_tendsto_of_lt | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Dense",
"DenselyOrdered",
"FirstCountableTopology",
"Set",
"StrictAnti"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α]
[FirstCountableTopology α] {s : Set α} (hs : Dense s) (x : α) :
∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ (Ioi x ∩ s)) ∧ Tendsto u atTop (𝓝 x) | hs.exists_seq_strictMono_tendsto (α := αᵒᵈ) x | theorem | Dense.exists_seq_strictAnti_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"Dense",
"DenselyOrdered",
"FirstCountableTopology",
"NoMaxOrder",
"Set",
"StrictAnti"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
DenseRange.exists_seq_strictAnti_tendsto_of_lt {β : Type*} [LinearOrder β]
[DenselyOrdered α] [FirstCountableTopology α] {f : β → α} {x y : α} (hf : DenseRange f)
(hmono : Monotone f) (hlt : x < y) :
∃ u : ℕ → β, StrictAnti u ∧ (∀ n, f (u n) ∈ Ioo x y) ∧ Tendsto (f ∘ u) atTop (𝓝 x) | by
simpa using! hf.exists_seq_strictMono_tendsto_of_lt (α := αᵒᵈ) (β := βᵒᵈ) hmono.dual
(OrderDual.toDual_lt_toDual.2 hlt) | theorem | DenseRange.exists_seq_strictAnti_tendsto_of_lt | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"DenseRange",
"DenselyOrdered",
"FirstCountableTopology",
"LinearOrder",
"Monotone",
"StrictAnti"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
DenseRange.exists_seq_strictAnti_tendsto {β : Type*} [LinearOrder β] [DenselyOrdered α]
[NoMaxOrder α] [FirstCountableTopology α] {f : β → α} (hf : DenseRange f) (hmono : Monotone f)
(x : α) :
∃ u : ℕ → β, StrictAnti u ∧ (∀ n, f (u n) ∈ Ioi x) ∧ Tendsto (f ∘ u) atTop (𝓝 x) | hf.exists_seq_strictMono_tendsto (α := αᵒᵈ) (β := βᵒᵈ) hmono.dual x | theorem | DenseRange.exists_seq_strictAnti_tendsto | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"DenseRange",
"DenselyOrdered",
"FirstCountableTopology",
"LinearOrder",
"Monotone",
"NoMaxOrder",
"StrictAnti"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_le_const_iff_forall_gt_eventually_lt_const [FirstCountableTopology α]
{l : Filter γ} [CountableInterFilter l] {f : γ → α} {a : α} :
(∀ᶠ x in l, f x ≤ a) ↔ ∀ b, a < b → ∀ᶠ x in l, f x < b | where
mp h c hbc := h.mono <| fun x hx ↦ lt_of_le_of_lt hx hbc
mpr h := by
rcases exists_glb_Ioi a with ⟨d, hd⟩
obtain rfl | H0 := glb_Ioi_eq_self_or_Ioi_eq_Ici _ hd
· obtain h | _ := isTop_or_exists_gt d
· exact .of_forall (fun _ ↦ h _)
obtain ⟨u, -, -, hu_tt, hu_gt⟩ := hd.exists_seq_antito... | theorem | eventually_le_const_iff_forall_gt_eventually_lt_const | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"CountableInterFilter",
"Filter",
"FirstCountableTopology",
"Set.compl_Iic",
"Set.compl_Iio",
"Set.mem_Iic",
"Set.mem_Iio",
"compl_inj_iff",
"eventually_countable_forall",
"isTop_or_exists_gt",
"le_of_lt",
"lt_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_const_le_iff_forall_lt_eventually_const_lt [FirstCountableTopology α]
{l : Filter γ} [CountableInterFilter l] {f : γ → α} {a : α} :
(∀ᶠ x in l, a ≤ f x) ↔ ∀ b, b < a → ∀ᶠ x in l, b < f x | eventually_le_const_iff_forall_gt_eventually_lt_const (α := αᵒᵈ) | theorem | eventually_const_le_iff_forall_lt_eventually_const_lt | Topology.Order | Mathlib/Topology/Order/IsLUB.lean | [] | [
"CountableInterFilter",
"Filter",
"FirstCountableTopology",
"eventually_le_const_iff_forall_gt_eventually_lt_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsNormal.continuous {f : α → β} (hf : IsNormal f) : Continuous f | by
rw [OrderTopology.continuous_iff]
refine fun b ↦ ⟨?_, ((isLowerSet_Iio b).preimage hf.strictMono.monotone).isOpen⟩
rw [← isClosed_compl_iff, ← Set.preimage_compl, Set.compl_Ioi]
obtain ha | ⟨a, ha⟩ := ((isLowerSet_Iic b).preimage hf.strictMono.monotone).eq_univ_or_Iio
· exact ha ▸ isClosed_univ
· obtain ... | theorem | Order.IsNormal.continuous | Topology.Order | Mathlib/Topology/Order/IsNormal.lean | [] | [
"Continuous",
"OrderTopology.continuous_iff",
"Set.preimage_compl",
"bddAbove_Iio",
"isClosed_Iic",
"isClosed_compl_iff",
"isClosed_empty",
"isClosed_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isNormal_iff_strictMono_and_continuous {f : α → β} :
IsNormal f ↔ StrictMono f ∧ Continuous f | where
mp hf := ⟨hf.strictMono, hf.continuous⟩
mpr := by
rintro ⟨hs, hc⟩
refine ⟨hs, fun {a} ha ↦ (isLUB_of_mem_closure ?_ ?_).2⟩
· rintro _ ⟨b, hb, rfl⟩
exact (hs hb).le
· apply image_closure_subset_closure_image hc (mem_image_of_mem ..)
exact ha.isLUB_Iio.mem_closure (Iio_nonempty.2 ha.... | theorem | Order.isNormal_iff_strictMono_and_continuous | Topology.Order | Mathlib/Topology/Order/IsNormal.lean | [] | [
"Continuous",
"StrictMono",
"image_closure_subset_closure_image",
"isLUB_of_mem_closure"
] | A normal function between well-orders is equivalent to a strictly monotone,
continuous function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousInf (L : Type*) [TopologicalSpace L] [Min L] : Prop where
/-- The infimum is continuous -/
continuous_inf : Continuous fun p : L × L => p.1 ⊓ p.2 | class | ContinuousInf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"TopologicalSpace",
"continuous_inf"
] | Let `L` be a topological space and let `L×L` be equipped with the product topology and let
`⊓:L×L → L` be an infimum. Then `L` is said to have *(jointly) continuous infimum* if the map
`⊓:L×L → L` is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousSup (L : Type*) [TopologicalSpace L] [Max L] : Prop where
/-- The supremum is continuous -/
continuous_sup : Continuous fun p : L × L => p.1 ⊔ p.2 | class | ContinuousSup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"TopologicalSpace",
"continuous_sup"
] | Let `L` be a topological space and let `L×L` be equipped with the product topology and let
`⊓:L×L → L` be a supremum. Then `L` is said to have *(jointly) continuous supremum* if the map
`⊓:L×L → L` is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
OrderDual.continuousSup (L : Type*) [TopologicalSpace L] [Min L]
[h : ContinuousInf L] : ContinuousSup Lᵒᵈ | where
continuous_sup := h.continuous_inf | instance | OrderDual.continuousSup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousInf",
"ContinuousSup",
"TopologicalSpace",
"continuous_sup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
OrderDual.continuousInf (L : Type*) [TopologicalSpace L] [Max L]
[h : ContinuousSup L] : ContinuousInf Lᵒᵈ | where
continuous_inf := h.continuous_sup | instance | OrderDual.continuousInf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousInf",
"ContinuousSup",
"TopologicalSpace",
"continuous_inf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TopologicalLattice (L : Type*) [TopologicalSpace L] [Lattice L] : Prop
extends ContinuousInf L, ContinuousSup L | class | TopologicalLattice | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousInf",
"ContinuousSup",
"Lattice",
"TopologicalSpace"
] | Let `L` be a lattice equipped with a topology such that `L` has continuous infimum and supremum.
Then `L` is said to be a *topological lattice*. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
OrderDual.topologicalLattice (L : Type*) [TopologicalSpace L]
[Lattice L] [TopologicalLattice L] : TopologicalLattice Lᵒᵈ | where | instance | OrderDual.topologicalLattice | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Lattice",
"TopologicalLattice",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_inf [Min L] [ContinuousInf L] : Continuous fun p : L × L => p.1 ⊓ p.2 | ContinuousInf.continuous_inf | theorem | continuous_inf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.inf [Min L] [ContinuousInf L] {f g : X → L} (hf : Continuous f)
(hg : Continuous g) : Continuous fun x => f x ⊓ g x | continuous_inf.comp (hf.prodMk hg :) | theorem | Continuous.inf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_sup [Max L] [ContinuousSup L] : Continuous fun p : L × L => p.1 ⊔ p.2 | ContinuousSup.continuous_sup | theorem | continuous_sup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.sup [Max L] [ContinuousSup L] {f g : X → L} (hf : Continuous f)
(hg : Continuous g) : Continuous fun x => f x ⊔ g x | continuous_sup.comp (hf.prodMk hg :) | theorem | Continuous.sup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous",
"ContinuousSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sup_nhds' [Max L] [ContinuousSup L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
Tendsto (f ⊔ g) l (𝓝 (x ⊔ y)) | (continuous_sup.tendsto _).comp (hf.prodMk_nhds hg) | lemma | Filter.Tendsto.sup_nhds' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sup_nhds [Max L] [ContinuousSup L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
Tendsto (fun i => f i ⊔ g i) l (𝓝 (x ⊔ y)) | hf.sup_nhds' hg | lemma | Filter.Tendsto.sup_nhds | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inf_nhds' [Min L] [ContinuousInf L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
Tendsto (f ⊓ g) l (𝓝 (x ⊓ y)) | (continuous_inf.tendsto _).comp (hf.prodMk_nhds hg) | lemma | Filter.Tendsto.inf_nhds' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inf_nhds [Min L] [ContinuousInf L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) :
Tendsto (fun i => f i ⊓ g i) l (𝓝 (x ⊓ y)) | hf.inf_nhds' hg | lemma | Filter.Tendsto.inf_nhds | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finset_sup'_nhds [SemilatticeSup L] [ContinuousSup L]
(hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
Tendsto (s.sup' hne f) l (𝓝 (s.sup' hne g)) | by
induction hne using Finset.Nonempty.cons_induction with
| singleton => simpa using hs
| cons a s ha hne ihs =>
rw [forall_mem_cons] at hs
simp only [sup'_cons, hne]
exact hs.1.sup_nhds (ihs hs.2) | lemma | Filter.Tendsto.finset_sup'_nhds | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousSup",
"Finset.Nonempty.cons_induction",
"SemilatticeSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finset_sup'_nhds_apply [SemilatticeSup L] [ContinuousSup L]
(hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
Tendsto (fun a ↦ s.sup' hne (f · a)) l (𝓝 (s.sup' hne g)) | by
simpa only [← Finset.sup'_apply] using finset_sup'_nhds hne hs | lemma | Filter.Tendsto.finset_sup'_nhds_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousSup",
"Finset.sup'_apply",
"SemilatticeSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finset_inf'_nhds [SemilatticeInf L] [ContinuousInf L]
(hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
Tendsto (s.inf' hne f) l (𝓝 (s.inf' hne g)) | finset_sup'_nhds (L := Lᵒᵈ) hne hs | lemma | Filter.Tendsto.finset_inf'_nhds | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousInf",
"SemilatticeInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finset_inf'_nhds_apply [SemilatticeInf L] [ContinuousInf L]
(hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
Tendsto (fun a ↦ s.inf' hne (f · a)) l (𝓝 (s.inf' hne g)) | finset_sup'_nhds_apply (L := Lᵒᵈ) hne hs | lemma | Filter.Tendsto.finset_inf'_nhds_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousInf",
"SemilatticeInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finset_sup_nhds [SemilatticeSup L] [OrderBot L] [ContinuousSup L]
(hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (s.sup f) l (𝓝 (s.sup g)) | by
rcases s.eq_empty_or_nonempty with rfl | hne
· simpa using! tendsto_const_nhds
· simp only [← sup'_eq_sup hne]
exact finset_sup'_nhds hne hs | lemma | Filter.Tendsto.finset_sup_nhds | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousSup",
"OrderBot",
"SemilatticeSup",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finset_sup_nhds_apply [SemilatticeSup L] [OrderBot L] [ContinuousSup L]
(hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
Tendsto (fun a ↦ s.sup (f · a)) l (𝓝 (s.sup g)) | by
simpa only [← Finset.sup_apply] using finset_sup_nhds hs | lemma | Filter.Tendsto.finset_sup_nhds_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousSup",
"Finset.sup_apply",
"OrderBot",
"SemilatticeSup"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finset_inf_nhds [SemilatticeInf L] [OrderTop L] [ContinuousInf L]
(hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (s.inf f) l (𝓝 (s.inf g)) | finset_sup_nhds (L := Lᵒᵈ) hs | lemma | Filter.Tendsto.finset_inf_nhds | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousInf",
"OrderTop",
"SemilatticeInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finset_inf_nhds_apply [SemilatticeInf L] [OrderTop L] [ContinuousInf L]
(hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) :
Tendsto (fun a ↦ s.inf (f · a)) l (𝓝 (s.inf g)) | finset_sup_nhds_apply (L := Lᵒᵈ) hs | lemma | Filter.Tendsto.finset_inf_nhds_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousInf",
"OrderTop",
"SemilatticeInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.sup' (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (f ⊔ g) x | hf.sup_nhds' hg | lemma | ContinuousAt.sup' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.sup (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (fun a ↦ f a ⊔ g a) x | hf.sup' hg | lemma | ContinuousAt.sup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.sup' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (f ⊔ g) s x | hf.sup_nhds' hg | lemma | ContinuousWithinAt.sup' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.sup (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (fun a ↦ f a ⊔ g a) s x | hf.sup' hg | lemma | ContinuousWithinAt.sup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.sup' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (f ⊔ g) s | fun x hx ↦
(hf x hx).sup' (hg x hx) | lemma | ContinuousOn.sup' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.sup (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun a ↦ f a ⊔ g a) s | hf.sup' hg | lemma | ContinuousOn.sup | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.sup' (hf : Continuous f) (hg : Continuous g) : Continuous (f ⊔ g) | hf.sup hg | lemma | Continuous.sup' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.inf' (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (f ⊓ g) x | hf.inf_nhds' hg | lemma | ContinuousAt.inf' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.inf (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (fun a ↦ f a ⊓ g a) x | hf.inf' hg | lemma | ContinuousAt.inf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.inf' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (f ⊓ g) s x | hf.inf_nhds' hg | lemma | ContinuousWithinAt.inf' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousWithinAt.inf (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (fun a ↦ f a ⊓ g a) s x | hf.inf' hg | lemma | ContinuousWithinAt.inf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.inf' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (f ⊓ g) s | fun x hx ↦
(hf x hx).inf' (hg x hx) | lemma | ContinuousOn.inf' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.inf (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun a ↦ f a ⊓ g a) s | hf.inf' hg | lemma | ContinuousOn.inf | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Continuous.inf' (hf : Continuous f) (hg : Continuous g) : Continuous (f ⊓ g) | hf.inf hg | lemma | Continuous.inf' | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousAt.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
ContinuousAt (fun a ↦ s.sup' hne (f · a)) x | Tendsto.finset_sup'_nhds_apply hne hs | lemma | ContinuousAt.finset_sup'_apply | Topology.Order | Mathlib/Topology/Order/Lattice.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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