state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case inr
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen a
t : Set α := {x | x ∈ s... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h') | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } := by
nontriviality α
let s := { x : α | ∃ y, x ⋖ y }
have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id
cho... | Mathlib.Topology.Order.Basic.1338_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } | Mathlib_Topology_Order_Basic |
case H
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen a
t : Set α := {x | x ∈ s ∧... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' this.countable_of_isOpen (fun x hx => _) fun x hx => ⟨x, hz x hx, le_rfl⟩ | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } := by
nontriviality α
let s := { x : α | ∃ y, x ⋖ y }
have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id
cho... | Mathlib.Topology.Order.Basic.1338_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } | Mathlib_Topology_Order_Basic |
case H
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen a
t : Set α := {x | x ∈ s ∧... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | suffices H : Ioc (z x) x = Ioo (z x) (y x) | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } := by
nontriviality α
let s := { x : α | ∃ y, x ⋖ y }
have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id
cho... | Mathlib.Topology.Order.Basic.1338_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } | Mathlib_Topology_Order_Basic |
case H
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen a
t : Set α := {x | x ∈ s ∧... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [H] | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } := by
nontriviality α
let s := { x : α | ∃ y, x ⋖ y }
have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id
cho... | Mathlib.Topology.Order.Basic.1338_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } | Mathlib_Topology_Order_Basic |
case H
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen a
t : Set α := {x | x ∈ s ∧... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact isOpen_Ioo | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } := by
nontriviality α
let s := { x : α | ∃ y, x ⋖ y }
have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id
cho... | Mathlib.Topology.Order.Basic.1338_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } | Mathlib_Topology_Order_Basic |
case H
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen a
t : Set α := {x | x ∈ s ∧... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩ | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } := by
nontriviality α
let s := { x : α | ∃ y, x ⋖ y }
have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id
cho... | Mathlib.Topology.Order.Basic.1338_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_covby_right [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
⊢ Set.Countable {x | ∃ y, x < y ∧ Ioo x y = ∅} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_right | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
@[deprecated countable_setOf_covby_right]
theorem countable_of_isolated_right' [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } := by
| Mathlib.Topology.Order.Basic.1380_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
@[deprecated countable_setOf_covby_right]
theorem countable_of_isolated_right' [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x < y ∧ Ioo x y = ∅ } | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
⊢ Set.Countable {x | ∃ y, y ⋖ x} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | convert countable_setOf_covby_right (α := αᵒᵈ) using 5 | /-- The set of points which are isolated on the left is countable when the space is
second-countable. -/
theorem countable_setOf_covby_left [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, y ⋖ x } := by
| Mathlib.Topology.Order.Basic.1388_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the left is countable when the space is
second-countable. -/
theorem countable_setOf_covby_left [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, y ⋖ x } | Mathlib_Topology_Order_Basic |
case h.e'_2.h.h.e'_2.h.h.h.e'_2.h.h.a
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
e_1✝ : α = αᵒᵈ
x✝¹ x✝ : α
⊢ x✝ ⋖ x✝¹ ↔ x✝¹ ⋖ x✝ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact toDual_covby_toDual_iff.symm | /-- The set of points which are isolated on the left is countable when the space is
second-countable. -/
theorem countable_setOf_covby_left [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, y ⋖ x } := by
convert countable_setOf_covby_right (α := αᵒᵈ) using 5
| Mathlib.Topology.Order.Basic.1388_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the left is countable when the space is
second-countable. -/
theorem countable_setOf_covby_left [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, y ⋖ x } | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
⊢ Set.Countable {x | ∃ y < x, Ioo y x = ∅} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simpa only [← covby_iff_Ioo_eq] using countable_setOf_covby_left | /-- The set of points which are isolated on the left is countable when the space is
second-countable. -/
theorem countable_of_isolated_left' [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by
| Mathlib.Topology.Order.Basic.1395_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the left is countable when the space is
second-countable. -/
theorem countable_of_isolated_left' [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
⊢ Set.Countable {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | nontriviality β | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
⊢ Set.Countable {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have : Nonempty α := Nonempty.map f (by infer_instance) | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
⊢ Nonempty β | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | infer_instance | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
⊢ Set.Countable {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | let s := {x | ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
⊢ Set.Countable {x | ∃ z, f x < z ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y := fun x hx ↦ hx | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this✝ : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
this : ∀ x ∈ s, ∃ z, f x < z ∧ ∀ (y : β), x < y... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | choose! z hz using this | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have I : InjOn f s := by
apply StrictMonoOn.injOn
intro x hx y _ hxy
calc
f x < z x := (hz x hx).1
_ ≤ f y := (hz x hx).2 y hxy | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply StrictMonoOn.injOn | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
case H
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | intro x hx y _ hxy | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
case H
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | calc
f x < z x := (hz x hx).1
_ ≤ f y := (hz x hx).2 y hxy | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have fs_count : (f '' s).Countable := by
have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by
rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv
wlog hle : u ≤ v generalizing u v
· exact (this v vs u us huv.symm (le_of_not_le hle)).symm
have hlt : u < v := hle.lt_of_ne (ne_of_a... | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by
rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv
wlog hle : u ≤ v generalizing u v
· exact (this v vs u us huv.symm (le_of_not_le hle)).symm
have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)
apply disjoint_iff_foral... | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
case intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | wlog hle : u ≤ v generalizing u v | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
case intro.intro.intro.intro.inr
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this✝ : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz :... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact (this v vs u us huv.symm (le_of_not_le hle)).symm | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv) | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply disjoint_iff_forall_ne.2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rintro a ha b hb rfl | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1) | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply Set.PairwiseDisjoint.countable_of_Ioo A | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rintro _ ⟨y, ys, rfl⟩ | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
case intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : β → α
inst✝ : SecondCountableTopology α
✝ : Nontrivial β
this : Nonempty α
s : Set β := {x | ∃ z, f x < z ∧ ∀ (y : β), x < y → z ≤ f y}
z : β → α
hz : ∀ x ∈ s, f x < z x ∧ ∀ (y : β), x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib.Topology.Order.Basic.1414_0.Npdof1X5b8sCkZ2 | /-- For a function taking values in a second countable space, the set of points `x` for
which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/
theorem countable_image_lt_image_Ioi [LinearOrder β] (f : β → α) [SecondCountableTopology α] :
Set.Countable {x | ∃ z, f x < z ∧ ∀ y, x < y → z... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
⊢ IsCountablyGenerated atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | by_cases h : ∃ (x : α), IsTop x | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
| Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case pos
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ∃ x, IsTop x
⊢ IsCountablyGenerated atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases h with ⟨x, hx⟩ | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case pos.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
x : α
hx : IsTop x
⊢ IsCountablyGenerated atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [atTop_eq_pure_of_isTop hx] | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
| Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case pos.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
x : α
hx : IsTop x
⊢ IsCountablyGenerated (pure x) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact isCountablyGenerated_pure x | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
| Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case neg
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
⊢ IsCountablyGenerated atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩ | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case neg.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
⊢ IsCountablyGenerated atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have : Countable b := by exact Iff.mpr countable_coe_iff b_count | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
⊢ Countable ↑b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact Iff.mpr countable_coe_iff b_count | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case neg.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
⊢ IsCountablyGenerated atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
intro s
have : (s : Set α) ≠ ∅ := by
intro H
apply b_ne
convert s.2
exact H.symm
exact Iff.mp nmem_singleton_empty this | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
⊢ ∀ (s : ↑b), ∃ x, x ∈ ↑s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | intro s | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
s : ↑b
⊢ ∃ x, x ∈ ↑s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have : (s : Set α) ≠ ∅ := by
intro H
apply b_ne
convert s.2
exact H.symm | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
s : ↑b
⊢ ↑s ≠ ∅ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | intro H | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
s : ↑b
H : ↑s = ∅
⊢ False | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply b_ne | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
s : ↑b
H : ↑s = ∅
⊢ ∅ ∈ b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | convert s.2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case h.e'_4
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
s : ↑b
H : ↑s = ∅
⊢ ∅ = ↑s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact H.symm | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Countable ↑b
s : ↑b
this : ↑s ≠ ∅
⊢ ∃ x, x ∈ ↑s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact Iff.mp nmem_singleton_empty this | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case neg.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
A : ∀ (s : ↑b), ∃ x, x ∈ ↑s
⊢ I... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | choose a ha using A | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case neg.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have : (atTop : Filter α) = (generate (Ici '' (range a))) := by
apply atTop_eq_generate_of_not_bddAbove
intro ⟨x, hx⟩
simp only [IsTop, not_exists, not_forall, not_le] at h
rcases h x with ⟨y, hy⟩
obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=
hb.exists_subset_of_mem_ope... | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
⊢ atTop = generate (I... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | apply atTop_eq_generate_of_not_bddAbove | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case hs
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
⊢ ¬BddAbove (... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | intro ⟨x, hx⟩ | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case hs
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
hx : x ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp only [IsTop, not_exists, not_forall, not_le] at h | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case hs
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
hx : x ∈ upperBounds (ran... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases h x with ⟨y, hy⟩ | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case hs.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
hx : x ∈ upperBound... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=
hb.exists_subset_of_mem_open hy isOpen_Ioi | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case hs.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
h... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _) | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case hs.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
h... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩) | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case hs.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
h... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact lt_irrefl _ (I.trans_lt J) | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case neg.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [this] | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
case neg.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact ⟨_, (countable_range _).image _, rfl⟩ | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) := by
by_cases h : ∃ (x : α), IsTop x
· rcases h with ⟨x, hx⟩
rw [atTop_eq_pure_of_isTop hx]
exact isCountablyGenerated_pure x
· rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
... | Mathlib.Topology.Order.Basic.1468_0.Npdof1X5b8sCkZ2 | instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
IsCountablyGenerated (atTop : Filter α) | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : LinearOrder α
inst✝⁵ : OrderTopology α
ι : Type u_1
π : ι → Type u_2
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → LinearOrder (π i)
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : ∀ (i : ι), OrderTopology (π i)
a b x : (i : ι) → π i
a' b' x' : ι → α
inst✝... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a) | theorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by
| Mathlib.Topology.Order.Basic.1541_0.Npdof1X5b8sCkZ2 | theorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : LinearOrder α
inst✝⁵ : OrderTopology α
ι : Type u_1
π : ι → Type u_2
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → LinearOrder (π i)
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : ∀ (i : ι), OrderTopology (π i)
a b x : (i : ι) → π i
a' b' x' : ι → α
inst✝... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact Iio_mem_nhds (ha i) | theorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := by
refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Iio_subset a)
| Mathlib.Topology.Order.Basic.1541_0.Npdof1X5b8sCkZ2 | theorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : LinearOrder α
inst✝⁵ : OrderTopology α
ι : Type u_1
π : ι → Type u_2
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → LinearOrder (π i)
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : ∀ (i : ι), OrderTopology (π i)
a b x : (i : ι) → π i
a' b' x' : ι → α
inst✝... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b) | theorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by
| Mathlib.Topology.Order.Basic.1558_0.Npdof1X5b8sCkZ2 | theorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : LinearOrder α
inst✝⁵ : OrderTopology α
ι : Type u_1
π : ι → Type u_2
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → LinearOrder (π i)
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : ∀ (i : ι), OrderTopology (π i)
a b x : (i : ι) → π i
a' b' x' : ι → α
inst✝... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact Ioc_mem_nhds (ha i) (hb i) | theorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by
refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioc_subset a b)
| Mathlib.Topology.Order.Basic.1558_0.Npdof1X5b8sCkZ2 | theorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : LinearOrder α
inst✝⁵ : OrderTopology α
ι : Type u_1
π : ι → Type u_2
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → LinearOrder (π i)
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : ∀ (i : ι), OrderTopology (π i)
a b x : (i : ι) → π i
a' b' x' : ι → α
inst✝... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b) | theorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by
| Mathlib.Topology.Order.Basic.1567_0.Npdof1X5b8sCkZ2 | theorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : LinearOrder α
inst✝⁵ : OrderTopology α
ι : Type u_1
π : ι → Type u_2
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → LinearOrder (π i)
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : ∀ (i : ι), OrderTopology (π i)
a b x : (i : ι) → π i
a' b' x' : ι → α
inst✝... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact Ico_mem_nhds (ha i) (hb i) | theorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by
refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ico_subset a b)
| Mathlib.Topology.Order.Basic.1567_0.Npdof1X5b8sCkZ2 | theorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : LinearOrder α
inst✝⁵ : OrderTopology α
ι : Type u_1
π : ι → Type u_2
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → LinearOrder (π i)
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : ∀ (i : ι), OrderTopology (π i)
a b x : (i : ι) → π i
a' b' x' : ι → α
inst✝... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b) | theorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by
| Mathlib.Topology.Order.Basic.1576_0.Npdof1X5b8sCkZ2 | theorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : LinearOrder α
inst✝⁵ : OrderTopology α
ι : Type u_1
π : ι → Type u_2
inst✝⁴ : Finite ι
inst✝³ : (i : ι) → LinearOrder (π i)
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : ∀ (i : ι), OrderTopology (π i)
a b x : (i : ι) → π i
a' b' x' : ι → α
inst✝... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact Ioo_mem_nhds (ha i) (hb i) | theorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by
refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)
| Mathlib.Topology.Order.Basic.1576_0.Npdof1X5b8sCkZ2 | theorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : NoMaxOrder α
x : α
⊢ Disjoint (𝓝 x) atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases exists_gt x with ⟨y, hy : x < y⟩ | theorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by
| Mathlib.Topology.Order.Basic.1587_0.Npdof1X5b8sCkZ2 | theorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop | Mathlib_Topology_Order_Basic |
case intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : NoMaxOrder α
x y : α
hy : x < y
⊢ Disjoint (𝓝 x) atTop | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y) | theorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by
rcases exists_gt x with ⟨y, hy : x < y⟩
| Mathlib.Topology.Order.Basic.1587_0.Npdof1X5b8sCkZ2 | theorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop | Mathlib_Topology_Order_Basic |
case intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : NoMaxOrder α
x y : α
hy : x < y
⊢ Disjoint (Iio y) {b | y ≤ b} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact disjoint_left.mpr fun z => not_le.2 | theorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop := by
rcases exists_gt x with ⟨y, hy : x < y⟩
refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)
| Mathlib.Topology.Order.Basic.1587_0.Npdof1X5b8sCkZ2 | theorem disjoint_nhds_atTop [NoMaxOrder α] (x : α) : Disjoint (𝓝 x) atTop | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
⊢ TFAE [s ∈ 𝓝[>] a, s ∈ 𝓝[Ioc a b] a, s ∈ 𝓝[Ioo a b] a, ∃ u ∈ Ioc a b, Ioo a u ⊆ s, ∃ u ∈ Ioi a, Ioo a u ⊆ s] | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | tfae_have 1 ↔ 2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_1_iff_2
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
⊢ s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab] | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
⊢ TFAE [s ∈ 𝓝[>] a, s ∈ 𝓝[Ioc a b] a, s ∈ 𝓝[Ioo a b] a, ∃ u ∈ Ioc a b, Ioo a u ⊆ s, ∃ u ∈ Ioi a, Ioo a u ⊆ s] | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | tfae_have 1 ↔ 3 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_1_iff_3
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
⊢ s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab] | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
⊢ TFAE [s ∈ 𝓝[>] a, s ∈ 𝓝[Ioc a b] a, s ∈ 𝓝[Ioo a b] a, ∃ u ∈ Ioc a b, Ioo a u ⊆ s, ∃... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | tfae_have 4 → 5 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_4_to_5
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
⊢ (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, Ioo a u ⊆ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩ | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, Ioo a u ⊆ s
⊢ TFAE [s ∈ 𝓝[>] ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | tfae_have 5 → 1 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_5_to_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, Ioo a u ⊆ s
⊢... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rintro ⟨u, hau, hu⟩ | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_5_to_1.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, I... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, Ioo a u ⊆ s
tfae_5_to_1 : (∃ u... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | tfae_have 1 → 4 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_1_to_4
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, Ioo a u ⊆ s
t... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | intro h | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_1_to_4
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, Ioo a u ⊆ s
t... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩ | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_1_to_4.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, I... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩ | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_1_to_4.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' ⟨u, au, fun x hx => _⟩ | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_1_to_4.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩ | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
case tfae_1_to_4.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact hx.1 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioc a b] a
tfae_1_iff_3 : s ∈ 𝓝[>] a ↔ s ∈ 𝓝[Ioo a b] a
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, Ioo a u ⊆ s
tfae_5_to_1 : (∃ u... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | tfae_finish | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib.Topology.Order.Basic.1634_0.Npdof1X5b8sCkZ2 | open List in
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nh... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
⊢ 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | by_cases ha : IsTop a | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by
| Mathlib.Topology.Order.Basic.1687_0.Npdof1X5b8sCkZ2 | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b | Mathlib_Topology_Order_Basic |
case pos
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
ha : IsTop a
⊢ 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp [ha, ha.isMax.Ioi_eq] | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by
by_cases ha : IsTop a
· | Mathlib.Topology.Order.Basic.1687_0.Npdof1X5b8sCkZ2 | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b | Mathlib_Topology_Order_Basic |
case neg
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
ha : ¬IsTop a
⊢ 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp only [ha, false_or] | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by
by_cases ha : IsTop a
· simp [ha, ha.isMax.Ioi_eq]
· | Mathlib.Topology.Order.Basic.1687_0.Npdof1X5b8sCkZ2 | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b | Mathlib_Topology_Order_Basic |
case neg
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
ha : ¬IsTop a
⊢ 𝓝[>] a = ⊥ ↔ ∃ b, a ⋖ b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [isTop_iff_isMax, not_isMax_iff] at ha | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by
by_cases ha : IsTop a
· simp [ha, ha.isMax.Ioi_eq]
· simp only [ha, false_or]
| Mathlib.Topology.Order.Basic.1687_0.Npdof1X5b8sCkZ2 | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b | Mathlib_Topology_Order_Basic |
case neg
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
ha : ∃ b, a < b
⊢ 𝓝[>] a = ⊥ ↔ ∃ b, a ⋖ b | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covby_iff_Ioo_eq] | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by
by_cases ha : IsTop a
· simp [ha, ha.isMax.Ioi_eq]
· simp only [ha, false_or]
rw [isTop_iff_isMax, not_isMax_iff] at ha
| Mathlib.Topology.Order.Basic.1687_0.Npdof1X5b8sCkZ2 | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
⊢ Set.Countable {x | 𝓝[>] x = ⊥} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or] | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_isolated_right [SecondCountableTopology α] :
{ x : α | 𝓝[>] x = ⊥ }.Countable := by
| Mathlib.Topology.Order.Basic.1702_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_isolated_right [SecondCountableTopology α] :
{ x : α | 𝓝[>] x = ⊥ }.Countable | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
⊢ Set.Countable ({a | IsTop a} ∪ {a | ∃ b, a ⋖ b}) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact (subsingleton_isTop α).countable.union countable_setOf_covby_right | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_isolated_right [SecondCountableTopology α] :
{ x : α | 𝓝[>] x = ⊥ }.Countable := by
simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]
| Mathlib.Topology.Order.Basic.1702_0.Npdof1X5b8sCkZ2 | /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_isolated_right [SecondCountableTopology α] :
{ x : α | 𝓝[>] x = ⊥ }.Countable | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : NoMaxOrder α
inst✝ : DenselyOrdered α
a : α
s : Set α
⊢ s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset] | /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by
| Mathlib.Topology.Order.Basic.1715_0.Npdof1X5b8sCkZ2 | /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : NoMaxOrder α
inst✝ : DenselyOrdered α
a : α
s : Set α
⊢ (∃ u ∈ Ioi a, Ioo a u ⊆ s) ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | constructor | /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by
rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]
... | Mathlib.Topology.Order.Basic.1715_0.Npdof1X5b8sCkZ2 | /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s | Mathlib_Topology_Order_Basic |
case mp
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : NoMaxOrder α
inst✝ : DenselyOrdered α
a : α
s : Set α
⊢ (∃ u ∈ Ioi a, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, Ioc a u ⊆ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rintro ⟨u, au, as⟩ | /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by
rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]
... | Mathlib.Topology.Order.Basic.1715_0.Npdof1X5b8sCkZ2 | /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s | Mathlib_Topology_Order_Basic |
case mp.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : NoMaxOrder α
inst✝ : DenselyOrdered α
a : α
s : Set α
u : α
au : u ∈ Ioi a
as : Ioo a u ⊆ s
⊢ ∃ u ∈ Ioi a, Ioc a u ⊆ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases exists_between au with ⟨v, hv⟩ | /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by
rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]
... | Mathlib.Topology.Order.Basic.1715_0.Npdof1X5b8sCkZ2 | /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s | Mathlib_Topology_Order_Basic |
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