fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
atBot_atTop_le_cocompact [NoMinOrder α] [NoMaxOrder α]
[OrderClosedTopology α] : atBot ⊔ atTop ≤ cocompact α :=
sup_le atBot_le_cocompact atTop_le_cocompact
@[simp 900] | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | atBot_atTop_le_cocompact | null |
cocompact_eq_atBot_atTop [NoMaxOrder α] [NoMinOrder α]
[OrderClosedTopology α] [CompactIccSpace α] : cocompact α = atBot ⊔ atTop :=
cocompact_le_atBot_atTop.antisymm atBot_atTop_le_cocompact
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | cocompact_eq_atBot_atTop | null |
cocompact_eq_atBot [NoMinOrder α] [OrderTop α]
[ClosedIicTopology α] [CompactIccSpace α] : cocompact α = atBot :=
cocompact_le_atBot.antisymm atBot_le_cocompact
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | cocompact_eq_atBot | null |
cocompact_eq_atTop [NoMaxOrder α] [OrderBot α]
[ClosedIciTopology α] [CompactIccSpace α] : cocompact α = atTop :=
cocompact_le_atTop.antisymm atTop_le_cocompact | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | cocompact_eq_atTop | null |
IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
(ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by
rcases (hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩
refine ⟨x, hxs, forall_mem_image.1 (fun _ hb => hx <| mem_image_of_mem f ?_)⟩
rwa [(image_id' s).symm] | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_isMinOn | The **extreme value theorem**: a continuous function realizes its minimum on a compact set. |
IsCompact.exists_forall_le' [ClosedIicTopology α] [NoMaxOrder α] {f : β → α}
{s : Set β} (hs : IsCompact s) (hf : ContinuousOn f s) {a : α} (hf' : ∀ b ∈ s, a < f b) :
∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b := by
rcases s.eq_empty_or_nonempty with (rfl | hs')
· obtain ⟨a', ha'⟩ := exists_gt a
exact ⟨a', ha', fun _ a ↦ a.elim⟩
· obtain ⟨x, hx, hx'⟩ := hs.exists_isMinOn hs' hf
exact ⟨f x, hf' x hx, hx'⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_forall_le' | If a continuous function lies strictly above `a` on a compact set,
it has a lower bound strictly above `a`. |
IsCompact.exists_isMaxOn [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
(ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMaxOn f s x :=
IsCompact.exists_isMinOn (α := αᵒᵈ) hs ne_s hf | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_isMaxOn | The **extreme value theorem**: a continuous function realizes its maximum on a compact set. |
ContinuousOn.exists_isMinOn' [ClosedIicTopology α] {s : Set β} {f : β → α}
(hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
(hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) : ∃ x ∈ s, IsMinOn f s x := by
rcases (hasBasis_cocompact.inf_principal _).eventually_iff.1 hc with ⟨K, hK, hKf⟩
have hsub : insert x₀ (K ∩ s) ⊆ s := insert_subset_iff.2 ⟨h₀, inter_subset_right⟩
obtain ⟨x, hx, hxf⟩ : ∃ x ∈ insert x₀ (K ∩ s), ∀ y ∈ insert x₀ (K ∩ s), f x ≤ f y :=
((hK.inter_right hsc).insert x₀).exists_isMinOn (insert_nonempty _ _) (hf.mono hsub)
refine ⟨x, hsub hx, fun y hy => ?_⟩
by_cases hyK : y ∈ K
exacts [hxf _ (Or.inr ⟨hyK, hy⟩), (hxf _ (Or.inl rfl)).trans (hKf ⟨hyK, hy⟩)] | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | ContinuousOn.exists_isMinOn' | The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
larger than a value in its image away from compact sets, then it has a minimum on this set. |
ContinuousOn.exists_isMaxOn' [ClosedIciTopology α] {s : Set β} {f : β → α}
(hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
(hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) : ∃ x ∈ s, IsMaxOn f s x :=
ContinuousOn.exists_isMinOn' (α := αᵒᵈ) hf hsc h₀ hc | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | ContinuousOn.exists_isMaxOn' | The **extreme value theorem**: if a function `f` is continuous on a closed set `s` and it is
smaller than a value in its image away from compact sets, then it has a maximum on this set. |
Continuous.exists_forall_le' [ClosedIicTopology α] {f : β → α} (hf : Continuous f)
(x₀ : β) (h : ∀ᶠ x in cocompact β, f x₀ ≤ f x) : ∃ x : β, ∀ y : β, f x ≤ f y :=
let ⟨x, _, hx⟩ := hf.continuousOn.exists_isMinOn' isClosed_univ (mem_univ x₀)
(by rwa [principal_univ, inf_top_eq])
⟨x, fun y => hx (mem_univ y)⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | Continuous.exists_forall_le' | The **extreme value theorem**: if a continuous function `f` is larger than a value in its range
away from compact sets, then it has a global minimum. |
Continuous.exists_forall_ge' [ClosedIciTopology α] {f : β → α} (hf : Continuous f)
(x₀ : β) (h : ∀ᶠ x in cocompact β, f x ≤ f x₀) : ∃ x : β, ∀ y : β, f y ≤ f x :=
Continuous.exists_forall_le' (α := αᵒᵈ) hf x₀ h | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | Continuous.exists_forall_ge' | The **extreme value theorem**: if a continuous function `f` is smaller than a value in its range
away from compact sets, then it has a global maximum. |
Continuous.exists_forall_le [ClosedIicTopology α] [Nonempty β] {f : β → α}
(hf : Continuous f) (hlim : Tendsto f (cocompact β) atTop) : ∃ x, ∀ y, f x ≤ f y := by
inhabit β
exact hf.exists_forall_le' default (hlim.eventually <| eventually_ge_atTop _) | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | Continuous.exists_forall_le | The **extreme value theorem**: if a continuous function `f` tends to infinity away from compact
sets, then it has a global minimum. |
Continuous.exists_forall_ge [ClosedIciTopology α] [Nonempty β] {f : β → α}
(hf : Continuous f) (hlim : Tendsto f (cocompact β) atBot) : ∃ x, ∀ y, f y ≤ f x :=
Continuous.exists_forall_le (α := αᵒᵈ) hf hlim | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | Continuous.exists_forall_ge | The **extreme value theorem**: if a continuous function `f` tends to negative infinity away from
compact sets, then it has a global maximum. |
@[to_additive /-- A continuous function with compact support has a global minimum. -/]
Continuous.exists_forall_le_of_hasCompactMulSupport [ClosedIicTopology α] [Nonempty β]
[One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
∃ x : β, ∀ y : β, f x ≤ f y := by
obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.isCompact_range hf).exists_isLeast (range_nonempty _)
rw [mem_lowerBounds, forall_mem_range] at hx
exact ⟨x, hx⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | Continuous.exists_forall_le_of_hasCompactMulSupport | A continuous function with compact support has a global minimum. |
@[to_additive /-- A continuous function with compact support has a global maximum. -/]
Continuous.exists_forall_ge_of_hasCompactMulSupport [ClosedIciTopology α] [Nonempty β]
[One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
∃ x : β, ∀ y : β, f y ≤ f x :=
Continuous.exists_forall_le_of_hasCompactMulSupport (α := αᵒᵈ) hf h | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | Continuous.exists_forall_ge_of_hasCompactMulSupport | A continuous function with compact support has a global maximum. |
IsCompact.bddBelow [ClosedIicTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) :
BddBelow s := by
rcases s.eq_empty_or_nonempty with rfl | hne
· exact bddBelow_empty
· obtain ⟨a, -, has⟩ := hs.exists_isLeast hne
exact ⟨a, has⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.bddBelow | A compact set is bounded below |
IsCompact.bddAbove [ClosedIciTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) :
BddAbove s :=
IsCompact.bddBelow (α := αᵒᵈ) hs | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.bddAbove | A compact set is bounded above |
IsCompact.bddBelow_image [ClosedIicTopology α] [Nonempty α] {f : β → α} {K : Set β}
(hK : IsCompact K) (hf : ContinuousOn f K) : BddBelow (f '' K) :=
(hK.image_of_continuousOn hf).bddBelow | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.bddBelow_image | A continuous function is bounded below on a compact set. |
IsCompact.bddAbove_image [ClosedIciTopology α] [Nonempty α] {f : β → α} {K : Set β}
(hK : IsCompact K) (hf : ContinuousOn f K) : BddAbove (f '' K) :=
IsCompact.bddBelow_image (α := αᵒᵈ) hK hf | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.bddAbove_image | A continuous function is bounded above on a compact set. |
@[to_additive /-- A continuous function with compact support is bounded below. -/]
Continuous.bddBelow_range_of_hasCompactMulSupport [ClosedIicTopology α] [One α]
{f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddBelow (range f) :=
(h.isCompact_range hf).bddBelow | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | Continuous.bddBelow_range_of_hasCompactMulSupport | A continuous function with compact support is bounded below. |
@[to_additive /-- A continuous function with compact support is bounded above. -/]
Continuous.bddAbove_range_of_hasCompactMulSupport [ClosedIciTopology α] [One α]
{f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddAbove (range f) :=
Continuous.bddBelow_range_of_hasCompactMulSupport (α := αᵒᵈ) hf h | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | Continuous.bddAbove_range_of_hasCompactMulSupport | A continuous function with compact support is bounded above. |
IsCompact.sSup_lt_iff_of_continuous [ClosedIciTopology α] {f : β → α} {K : Set β}
(hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y := by
refine ⟨fun h x hx => (le_csSup (hK.bddAbove_image hf) <| mem_image_of_mem f hx).trans_lt h,
fun h => ?_⟩
obtain ⟨x, hx, h2x⟩ := hK.exists_isMaxOn h0K hf
refine (csSup_le (h0K.image f) ?_).trans_lt (h x hx)
rintro _ ⟨x', hx', rfl⟩; exact h2x hx' | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.sSup_lt_iff_of_continuous | null |
IsCompact.lt_sInf_iff_of_continuous [ClosedIicTopology α] {f : β → α} {K : Set β}
(hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x :=
IsCompact.sSup_lt_iff_of_continuous (α := αᵒᵈ) hK h0K hf y | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.lt_sInf_iff_of_continuous | null |
IsCompact.sInf_mem [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : sInf s ∈ s :=
let ⟨_a, ha⟩ := hs.exists_isLeast ne_s
ha.csInf_mem | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.sInf_mem | null |
IsCompact.sSup_mem [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : sSup s ∈ s :=
IsCompact.sInf_mem (α := αᵒᵈ) hs ne_s | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.sSup_mem | null |
IsCompact.isGLB_sInf [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : IsGLB s (sInf s) :=
isGLB_csInf ne_s hs.bddBelow | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.isGLB_sInf | null |
IsCompact.isLUB_sSup [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : IsLUB s (sSup s) :=
IsCompact.isGLB_sInf (α := αᵒᵈ) hs ne_s | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.isLUB_sSup | null |
IsCompact.isLeast_sInf [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : IsLeast s (sInf s) :=
⟨hs.sInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.isLeast_sInf | null |
IsCompact.isGreatest_sSup [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : IsGreatest s (sSup s) :=
IsCompact.isLeast_sInf (α := αᵒᵈ) hs ne_s | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.isGreatest_sSup | null |
IsCompact.exists_sInf_image_eq_and_le [ClosedIicTopology α] {s : Set β}
(hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) :
∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).sInf_mem (ne_s.image f)
⟨x, hxs, hx.symm, fun _y hy =>
hx.trans_le <| csInf_le (hs.image_of_continuousOn hf).bddBelow <| mem_image_of_mem f hy⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_sInf_image_eq_and_le | null |
IsCompact.exists_sSup_image_eq_and_ge [ClosedIciTopology α] {s : Set β}
(hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) :
∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
IsCompact.exists_sInf_image_eq_and_le (α := αᵒᵈ) hs ne_s hf | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_sSup_image_eq_and_ge | null |
IsCompact.exists_sInf_image_eq [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
(ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x :=
let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf
⟨x, hxs, hx⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_sInf_image_eq | null |
IsCompact.exists_sSup_image_eq [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
(ne_s : s.Nonempty) : ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x :=
IsCompact.exists_sInf_image_eq (α := αᵒᵈ) hs ne_s | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_sSup_image_eq | null |
IsCompact.exists_isMinOn_mem_subset [ClosedIicTopology α] {f : β → α} {s t : Set β}
{z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
(hfz : ∀ z' ∈ t \ s, f z < f z') : ∃ x ∈ s, IsMinOn f t x :=
let ⟨x, hxt, hfx⟩ := ht.exists_isMinOn ⟨z, hz⟩ hf
⟨x, by_contra fun hxs => (hfz x ⟨hxt, hxs⟩).not_ge (hfx hz), hfx⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_isMinOn_mem_subset | null |
IsCompact.exists_isMaxOn_mem_subset [ClosedIciTopology α] {f : β → α} {s t : Set β}
{z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
(hfz : ∀ z' ∈ t \ s, f z' < f z) : ∃ x ∈ s, IsMaxOn f t x :=
let ⟨x, hxt, hfx⟩ := ht.exists_isMaxOn ⟨z, hz⟩ hf
⟨x, by_contra fun hxs => (hfz x ⟨hxt, hxs⟩).not_ge (hfx hz), hfx⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_isMaxOn_mem_subset | null |
IsCompact.exists_isLocalMin_mem_open [ClosedIicTopology α] {f : β → α} {s t : Set β}
{z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t)
(hfz : ∀ z' ∈ t \ s, f z < f z') (hs : IsOpen s) : ∃ x ∈ s, IsLocalMin f x :=
let ⟨x, hxs, h⟩ := ht.exists_isMinOn_mem_subset hf hz hfz
⟨x, hxs, h.isLocalMin <| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_isLocalMin_mem_open | null |
IsCompact.exists_isLocalMax_mem_open [ClosedIciTopology α] {f : β → α} {s t : Set β}
{z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t)
(hfz : ∀ z' ∈ t \ s, f z' < f z) (hs : IsOpen s) : ∃ x ∈ s, IsLocalMax f x :=
let ⟨x, hxs, h⟩ := ht.exists_isMaxOn_mem_subset hf hz hfz
⟨x, hxs, h.isLocalMax <| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.exists_isLocalMax_mem_open | null |
eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
s = Icc (sInf s) (sSup s) :=
eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.bddBelow h₂.bddAbove h₂.isClosed | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | eq_Icc_of_connected_compact | null |
IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCompact K)
(hf : Continuous ↿f) : Continuous fun x => sSup (f x '' K) := by
rcases eq_empty_or_nonempty K with (rfl | h0K)
· simp_rw [image_empty]
exact continuous_const
rw [continuous_iff_continuousAt]
intro x
obtain ⟨y, hyK, h2y, hy⟩ :=
hK.exists_sSup_image_eq_and_ge h0K
(show Continuous (f x) from hf.comp <| .prodMk_right x).continuousOn
rw [ContinuousAt, h2y, tendsto_order]
have := tendsto_order.mp ((show Continuous fun x => f x y
from hf.comp <| .prodMk_left _).tendsto x)
refine ⟨fun z hz => ?_, fun z hz => ?_⟩
· refine (this.1 z hz).mono fun x' hx' =>
hx'.trans_le <| le_csSup ?_ <| mem_image_of_mem (f x') hyK
exact hK.bddAbove_image (hf.comp <| .prodMk_right x').continuousOn
· have h : ({x} : Set γ) ×ˢ K ⊆ ↿f ⁻¹' Iio z := by
rintro ⟨x', y'⟩ ⟨(rfl : x' = x), hy'⟩
exact (hy y' hy').trans_lt hz
obtain ⟨u, v, hu, _, hxu, hKv, huv⟩ :=
generalized_tube_lemma isCompact_singleton hK (isOpen_Iio.preimage hf) h
refine eventually_of_mem (hu.mem_nhds (singleton_subset_iff.mp hxu)) fun x' hx' => ?_
rw [hK.sSup_lt_iff_of_continuous h0K
(show Continuous (f x') from hf.comp <| .prodMk_right x').continuousOn]
exact fun y' hy' => huv (mk_mem_prod hx' (hKv hy')) | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.continuous_sSup | If `f : γ → β → α` is a function that is continuous as a function on `γ × β`, `α` is a
conditionally complete linear order, and `K : Set β` is a compact set, then
`fun x ↦ sSup (f x '' K)` is a continuous function. -/
/- TODO: generalize. The following version seems to be true:
```
theorem IsCompact.tendsto_sSup {f : γ → β → α} {g : β → α} {K : Set β} {l : Filter γ}
(hK : IsCompact K) (hf : ∀ y ∈ K, Tendsto ↿f (l ×ˢ 𝓝[K] y) (𝓝 (g y)))
(hgc : ContinuousOn g K) :
Tendsto (fun x => sSup (f x '' K)) l (𝓝 (sSup (g '' K))) := _
```
Moreover, it seems that `hgc` follows from `hf` (Yury Kudryashov). |
IsCompact.continuous_sInf {f : γ → β → α} {K : Set β} (hK : IsCompact K)
(hf : Continuous ↿f) : Continuous fun x => sInf (f x '' K) :=
IsCompact.continuous_sSup (α := αᵒᵈ) hK hf | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | IsCompact.continuous_sInf | null |
image_Icc (hab : a ≤ b) (h : ContinuousOn f <| Icc a b) :
f '' Icc a b = Icc (sInf <| f '' Icc a b) (sSup <| f '' Icc a b) :=
eq_Icc_of_connected_compact ⟨(nonempty_Icc.2 hab).image f, isPreconnected_Icc.image f h⟩
(isCompact_Icc.image_of_continuousOn h) | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | image_Icc | null |
image_uIcc_eq_Icc (h : ContinuousOn f [[a, b]]) :
f '' [[a, b]] = Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) :=
image_Icc min_le_max h | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | image_uIcc_eq_Icc | null |
image_uIcc (h : ContinuousOn f <| [[a, b]]) :
f '' [[a, b]] = [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] := by
refine h.image_uIcc_eq_Icc.trans (uIcc_of_le ?_).symm
refine csInf_le_csSup ?_ ?_ (nonempty_uIcc.image _) <;> rw [h.image_uIcc_eq_Icc]
exacts [bddBelow_Icc, bddAbove_Icc] | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | image_uIcc | null |
sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
sInf (f '' Icc a b) ≤ f c := by
have := mem_image_of_mem f hc
rw [h.image_Icc (hc.1.trans hc.2)] at this
exact this.1 | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | sInf_image_Icc_le | null |
le_sSup_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
f c ≤ sSup (f '' Icc a b) := by
have := mem_image_of_mem f hc
rw [h.image_Icc (hc.1.trans hc.2)] at this
exact this.2 | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | le_sSup_image_Icc | null |
image_Icc_of_monotoneOn (hab : a ≤ b) (h : ContinuousOn f <| Icc a b)
(h' : MonotoneOn f <| Icc a b) : f '' Icc a b = Icc (f a) (f b) := by
rw [h.image_Icc hab]
congr!
· exact h'.sInf_image_Icc hab
· exact h'.sSup_image_Icc hab | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | image_Icc_of_monotoneOn | null |
image_Icc_of_antitoneOn (hab : a ≤ b) (h : ContinuousOn f <| Icc a b)
(h' : AntitoneOn f <| Icc a b) : f '' Icc a b = Icc (f b) (f a) := by
have : Icc (f b) (f a) = Icc (toDual (f a)) (toDual (f b)) := by rw [Icc_toDual]; rfl
rw [this]
exact image_Icc_of_monotoneOn (β := βᵒᵈ) hab h h'.dual_right | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Order.IsLUB",
"Mathlib.Topology.Order.LocalExtr"
] | Mathlib/Topology/Order/Compact.lean | image_Icc_of_antitoneOn | null |
range_Iio : HasCountableSeparatingOn X (· ∈ range Iio) s := by
constructor
rcases TopologicalSpace.exists_countable_dense X with ⟨s, hsc, hsd⟩
set t := s ∪ {x | ∃ y, y ⋖ x}
refine ⟨Iio '' t, .image ?_ _, ?_, ?_⟩
· exact hsc.union countable_setOf_covBy_left
· exact image_subset_range _ _
· rintro x - y - h
by_contra! hne
wlog hlt : x < y generalizing x y
· refine this y x ?_ hne.symm (hne.lt_or_gt.resolve_left hlt)
simpa only [iff_comm] using h
cases (Ioo x y).eq_empty_or_nonempty with
| inl he =>
specialize h (Iio y) (mem_image_of_mem _ (.inr ⟨x, hlt, by simpa using Set.ext_iff.mp he⟩))
simp [hlt.not_ge] at h
| inr hne =>
rcases hsd.inter_open_nonempty _ isOpen_Ioo hne with ⟨z, ⟨hxz, hzy⟩, hzs⟩
simpa [hxz, hzy.not_gt] using h (Iio z) (mem_image_of_mem _ (.inl hzs)) | instance | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.Filter.CountableSeparatingOn"
] | Mathlib/Topology/Order/CountableSeparating.lean | range_Iio | null |
range_Ioi : HasCountableSeparatingOn X (· ∈ range Ioi) s :=
.range_Iio (X := Xᵒᵈ) | instance | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.Filter.CountableSeparatingOn"
] | Mathlib/Topology/Order/CountableSeparating.lean | range_Ioi | null |
range_Iic : HasCountableSeparatingOn X (· ∈ range Iic) s :=
let ⟨t, htc, ht_sub, ht⟩ := (range_Ioi (X := X) (s := s)).1
⟨compl '' t, htc.image _, by simpa [← compl_inj_iff (x := Ioi _)] using ht_sub,
by simpa [not_iff_not]⟩ | instance | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.Filter.CountableSeparatingOn"
] | Mathlib/Topology/Order/CountableSeparating.lean | range_Iic | null |
range_Ici : HasCountableSeparatingOn X (· ∈ range Ici) s :=
range_Iic (X := Xᵒᵈ) | instance | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.Filter.CountableSeparatingOn"
] | Mathlib/Topology/Order/CountableSeparating.lean | range_Ici | null |
of_forall_eventually_lt_iff (h : ∀ x, ∀ᶠ a in l, f a < x ↔ g a < x) : f =ᶠ[l] g :=
of_forall_separating_preimage (· ∈ range Iio) <| forall_mem_range.2 <| fun x ↦ .set_eq (h x) | lemma | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.Filter.CountableSeparatingOn"
] | Mathlib/Topology/Order/CountableSeparating.lean | of_forall_eventually_lt_iff | null |
of_forall_eventually_le_iff (h : ∀ x, ∀ᶠ a in l, f a ≤ x ↔ g a ≤ x) : f =ᶠ[l] g :=
of_forall_separating_preimage (· ∈ range Iic) <| forall_mem_range.2 <| fun x ↦ .set_eq (h x) | lemma | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.Filter.CountableSeparatingOn"
] | Mathlib/Topology/Order/CountableSeparating.lean | of_forall_eventually_le_iff | null |
of_forall_eventually_gt_iff (h : ∀ x, ∀ᶠ a in l, x < f a ↔ x < g a) : f =ᶠ[l] g :=
of_forall_eventually_lt_iff (X := Xᵒᵈ) h | lemma | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.Filter.CountableSeparatingOn"
] | Mathlib/Topology/Order/CountableSeparating.lean | of_forall_eventually_gt_iff | null |
of_forall_eventually_ge_iff (h : ∀ x, ∀ᶠ a in l, x ≤ f a ↔ x ≤ g a) : f =ᶠ[l] g :=
of_forall_eventually_le_iff (X := Xᵒᵈ) h | lemma | Topology | [
"Mathlib.Topology.Order.Basic",
"Mathlib.Order.Filter.CountableSeparatingOn"
] | Mathlib/Topology/Order/CountableSeparating.lean | of_forall_eventually_ge_iff | null |
closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by
apply Subset.antisymm
· exact closure_minimal Ioi_subset_Ici_self isClosed_Ici
· rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]
exact isGLB_Ioi.mem_closure h | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | closure_Ioi' | The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. |
@[simp]
closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a :=
closure_Ioi' nonempty_Ioi | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | closure_Ioi | The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. |
closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a :=
closure_Ioi' (α := αᵒᵈ) h | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | closure_Iio' | The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom
element. |
@[simp]
closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a :=
closure_Iio' nonempty_Iio | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | closure_Iio | The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. |
@[simp]
closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· rcases hab.lt_or_gt with hab | hab
· rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]
have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab
simp only [insert_subset_iff, singleton_subset_iff]
exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩
· rw [Icc_eq_empty_of_lt hab]
exact empty_subset _ | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | closure_Ioo | The closure of the open interval `(a, b)` is the closed interval `[a, b]`. |
@[simp]
closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ioc_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)
rw [closure_Ioo hab] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | closure_Ioc | The closure of the interval `(a, b]` is the closed interval `[a, b]`. |
@[simp]
closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by
apply Subset.antisymm
· exact closure_minimal Ico_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)
rw [closure_Ioo hab]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | closure_Ico | The closure of the interval `[a, b)` is the closed interval `[a, b]`. |
interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by
rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | interior_Ici' | null |
interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=
interior_Ici' nonempty_Iio
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | interior_Ici | null |
interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a :=
interior_Ici' (α := αᵒᵈ) ha | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | interior_Iic' | null |
interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a :=
interior_Iic' nonempty_Ioi
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | interior_Iic | null |
interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by
rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | interior_Icc | null |
Icc_mem_nhds_iff [NoMinOrder α] [NoMaxOrder α] {a b x : α} :
Icc a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by
rw [← interior_Icc, mem_interior_iff_mem_nhds]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | Icc_mem_nhds_iff | null |
interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by
rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | interior_Ico | null |
Ico_mem_nhds_iff [NoMinOrder α] {a b x : α} : Ico a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by
rw [← interior_Ico, mem_interior_iff_mem_nhds]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | Ico_mem_nhds_iff | null |
interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by
rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | interior_Ioc | null |
Ioc_mem_nhds_iff [NoMaxOrder α] {a b x : α} : Ioc a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by
rw [← interior_Ioc, mem_interior_iff_mem_nhds] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | Ioc_mem_nhds_iff | null |
closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b :=
(closure_minimal interior_subset isClosed_Icc).antisymm <|
calc
Icc a b = closure (Ioo a b) := (closure_Ioo h).symm
_ ⊆ closure (interior (Icc a b)) :=
closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo) | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | closure_interior_Icc | null |
Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by
rcases eq_or_ne a b with (rfl | h)
· simp
· calc
Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self
_ = closure (Ioo a b) := (closure_Ioo h).symm
_ ⊆ closure (interior (Ioc a b)) :=
closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo) | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | Ioc_subset_closure_interior | null |
Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by
simpa only [Ioc_toDual] using
Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | Ico_subset_closure_interior | null |
frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by
simp [frontier, ha] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Ici' | null |
frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=
frontier_Ici' nonempty_Iio
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Ici | null |
frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by
simp [frontier, ha] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Iic' | null |
frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} :=
frontier_Iic' nonempty_Ioi
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Iic | null |
frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by
simp [frontier, closure_Ioi' ha] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Ioi' | null |
frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} :=
frontier_Ioi' nonempty_Ioi
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Ioi | null |
frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by
simp [frontier, closure_Iio' ha] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Iio' | null |
frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} :=
frontier_Iio' nonempty_Iio
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Iio | null |
frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) :
frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Icc | null |
frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by
rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Ioo | null |
frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by
rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Ico | null |
frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by
rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | frontier_Ioc | null |
nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) :
NeBot (𝓝[Ioi a] b) :=
mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Ioi' H₁] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | nhdsWithin_Ioi_neBot' | null |
nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) :=
nhdsWithin_Ioi_neBot' nonempty_Ioi H | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | nhdsWithin_Ioi_neBot | null |
nhdsGT_neBot_of_exists_gt {a : α} (H : ∃ b, a < b) : NeBot (𝓝[>] a) :=
nhdsWithin_Ioi_neBot' H (le_refl a) | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | nhdsGT_neBot_of_exists_gt | null |
nhdsGT_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) := nhdsWithin_Ioi_neBot le_rfl | instance | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | nhdsGT_neBot | null |
nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) :
NeBot (𝓝[Iio c] b) :=
mem_closure_iff_nhdsWithin_neBot.1 <| by rwa [closure_Iio' H₁] | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | nhdsWithin_Iio_neBot' | null |
nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) :=
nhdsWithin_Iio_neBot' nonempty_Iio H | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | nhdsWithin_Iio_neBot | null |
nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) :=
nhdsWithin_Iio_neBot' H (le_refl b) | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | nhdsWithin_Iio_self_neBot' | null |
nhdsLT_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) := nhdsWithin_Iio_neBot (le_refl a) | instance | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | nhdsLT_neBot | null |
right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) :=
(isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H) | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | right_nhdsWithin_Ico_neBot | null |
left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) :=
(isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H) | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | left_nhdsWithin_Ioc_neBot | null |
left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) :=
(isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H) | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | left_nhdsWithin_Ioo_neBot | null |
right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) :=
(isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H) | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | right_nhdsWithin_Ioo_neBot | null |
comap_coe_nhdsLT_of_Ioo_subset (hb : s ⊆ Iio b) (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) :
comap ((↑) : s → α) (𝓝[<] b) = atTop := by
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· rintro ⟨t, ht, hts⟩
obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=
(mem_nhdsLT_iff_exists_mem_Ico_Ioo_subset h).mp ht
obtain ⟨y, hxy, hyb⟩ := exists_between hxb
refine mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) ?_
rintro ⟨z, hzs⟩ (hyz : y ≤ z)
exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩)
· intro hu
obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu
exact ⟨Ioo x b, Ioo_mem_nhdsLT (hb x.2), fun z hz => hx _ hz.1.le⟩ | theorem | Topology | [
"Mathlib.Topology.Order.IsLUB"
] | Mathlib/Topology/Order/DenselyOrdered.lean | comap_coe_nhdsLT_of_Ioo_subset | null |
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