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 |
|---|---|---|---|---|---|---|
C : Type u
inst✝³ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
inst✝¹ : HasEqualizer f g
inst✝ : HasEqualizer (G.map f) (G.map g)
Z : C
h : Z ⟶ X
w : h ≫ f = h ≫ g
⊢ G.map h ≫ G.map f = G.map h ≫ G.map g | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp only [← G.map_comp, w] | @[reassoc (attr := simp)]
theorem map_lift_equalizerComparison [HasEqualizer f g] [HasEqualizer (G.map f) (G.map g)] {Z : C}
{h : Z ⟶ X} (w : h ≫ f = h ≫ g) :
G.map (equalizer.lift h w) ≫ equalizerComparison f g G =
equalizer.lift (G.map h) (by | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1128_0.eJEUq2AFfmN187w | @[reassoc (attr | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝³ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
inst✝¹ : HasEqualizer f g
inst✝ : HasEqualizer (G.map f) (G.map g)
Z : C
h : Z ⟶ X
w : h ≫ f = h ≫ g
⊢ G.map (equalizer.lift h w) ≫ equalizerComparison f g G =
equalizer.lift (G.map h) (_ : G.map h ≫ G.map f... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | apply equalizer.hom_ext | @[reassoc (attr := simp)]
theorem map_lift_equalizerComparison [HasEqualizer f g] [HasEqualizer (G.map f) (G.map g)] {Z : C}
{h : Z ⟶ X} (w : h ≫ f = h ≫ g) :
G.map (equalizer.lift h w) ≫ equalizerComparison f g G =
equalizer.lift (G.map h) (by simp only [← G.map_comp, w]) := by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1128_0.eJEUq2AFfmN187w | @[reassoc (attr | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case h
C : Type u
inst✝³ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
inst✝¹ : HasEqualizer f g
inst✝ : HasEqualizer (G.map f) (G.map g)
Z : C
h : Z ⟶ X
w : h ≫ f = h ≫ g
⊢ (G.map (equalizer.lift h w) ≫ equalizerComparison f g G) ≫ equalizer.ι (G.map f) (G.map g) =
equa... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp [← G.map_comp] | @[reassoc (attr := simp)]
theorem map_lift_equalizerComparison [HasEqualizer f g] [HasEqualizer (G.map f) (G.map g)] {Z : C}
{h : Z ⟶ X} (w : h ≫ f = h ≫ g) :
G.map (equalizer.lift h w) ≫ equalizerComparison f g G =
equalizer.lift (G.map h) (by simp only [← G.map_comp, w]) := by
apply equalizer.hom_ext
... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1128_0.eJEUq2AFfmN187w | @[reassoc (attr | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝³ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
inst✝¹ : HasCoequalizer f g
inst✝ : HasCoequalizer (G.map f) (G.map g)
⊢ G.map f ≫ G.map (coequalizer.π f g) = G.map g ≫ G.map (coequalizer.π f g) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp only [← G.map_comp] | /-- The comparison morphism for the coequalizer of `f,g`. -/
noncomputable def coequalizerComparison [HasCoequalizer f g] [HasCoequalizer (G.map f) (G.map g)] :
coequalizer (G.map f) (G.map g) ⟶ G.obj (coequalizer f g) :=
coequalizer.desc (G.map (coequalizer.π _ _))
(by | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1137_0.eJEUq2AFfmN187w | /-- The comparison morphism for the coequalizer of `f,g`. -/
noncomputable def coequalizerComparison [HasCoequalizer f g] [HasCoequalizer (G.map f) (G.map g)] :
coequalizer (G.map f) (G.map g) ⟶ G.obj (coequalizer f g) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝³ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
inst✝¹ : HasCoequalizer f g
inst✝ : HasCoequalizer (G.map f) (G.map g)
⊢ G.map (f ≫ coequalizer.π f g) = G.map (g ≫ coequalizer.π f g) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rw [coequalizer.condition] | /-- The comparison morphism for the coequalizer of `f,g`. -/
noncomputable def coequalizerComparison [HasCoequalizer f g] [HasCoequalizer (G.map f) (G.map g)] :
coequalizer (G.map f) (G.map g) ⟶ G.obj (coequalizer f g) :=
coequalizer.desc (G.map (coequalizer.π _ _))
(by simp only [← G.map_comp]; | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1137_0.eJEUq2AFfmN187w | /-- The comparison morphism for the coequalizer of `f,g`. -/
noncomputable def coequalizerComparison [HasCoequalizer f g] [HasCoequalizer (G.map f) (G.map g)] :
coequalizer (G.map f) (G.map g) ⟶ G.obj (coequalizer f g) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝³ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
inst✝¹ : HasCoequalizer f g
inst✝ : HasCoequalizer (G.map f) (G.map g)
Z : C
h : Y ⟶ Z
w : f ≫ h = g ≫ h
⊢ G.map f ≫ G.map h = G.map g ≫ G.map h | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp only [← G.map_comp, w] | @[reassoc (attr := simp)]
theorem coequalizerComparison_map_desc [HasCoequalizer f g] [HasCoequalizer (G.map f) (G.map g)]
{Z : C} {h : Y ⟶ Z} (w : f ≫ h = g ≫ h) :
coequalizerComparison f g G ≫ G.map (coequalizer.desc h w) =
coequalizer.desc (G.map h) (by | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1150_0.eJEUq2AFfmN187w | @[reassoc (attr | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝³ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
inst✝¹ : HasCoequalizer f g
inst✝ : HasCoequalizer (G.map f) (G.map g)
Z : C
h : Y ⟶ Z
w : f ≫ h = g ≫ h
⊢ coequalizerComparison f g G ≫ G.map (coequalizer.desc h w) =
coequalizer.desc (G.map h) (_ : G.map f... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | apply coequalizer.hom_ext | @[reassoc (attr := simp)]
theorem coequalizerComparison_map_desc [HasCoequalizer f g] [HasCoequalizer (G.map f) (G.map g)]
{Z : C} {h : Y ⟶ Z} (w : f ≫ h = g ≫ h) :
coequalizerComparison f g G ≫ G.map (coequalizer.desc h w) =
coequalizer.desc (G.map h) (by simp only [← G.map_comp, w]) := by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1150_0.eJEUq2AFfmN187w | @[reassoc (attr | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case h
C : Type u
inst✝³ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
inst✝¹ : HasCoequalizer f g
inst✝ : HasCoequalizer (G.map f) (G.map g)
Z : C
h : Y ⟶ Z
w : f ≫ h = g ≫ h
⊢ coequalizer.π (G.map f) (G.map g) ≫ coequalizerComparison f g G ≫ G.map (coequalizer.desc h w) =
... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp [← G.map_comp] | @[reassoc (attr := simp)]
theorem coequalizerComparison_map_desc [HasCoequalizer f g] [HasCoequalizer (G.map f) (G.map g)]
{Z : C} {h : Y ⟶ Z} (w : f ≫ h = g ≫ h) :
coequalizerComparison f g G ≫ G.map (coequalizer.desc h w) =
coequalizer.desc (G.map h) (by simp only [← G.map_comp, w]) := by
apply coequa... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1150_0.eJEUq2AFfmN187w | @[reassoc (attr | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
inst✝ : IsSplitMono f
⊢ f ≫ 𝟙 Y = f ≫ retraction f ≫ f | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp | /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
Here we build the cone, and show in `isSplitMonoEqualizes` that it is a limit cone.
-/
-- @[simps (config := { rhsMd := semireducible })] Porting note: no semireducible
@[simps!]
noncomputable def coneOfIsSplitMono : Fork (𝟙 Y) (retraction f ≫ f) :=
F... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1190_0.eJEUq2AFfmN187w | /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
Here we build the cone, and show in `isSplitMonoEqualizes` that it is a limit cone.
-/
-- @[simps (config | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g : X✝ ⟶ Y✝
inst✝¹ : IsSplitMono f✝
X Y : C
f : X ⟶ Y
inst✝ : IsSplitMono f
s : Fork (𝟙 Y) (retraction f ≫ f)
⊢ (Fork.ι s ≫ retraction f) ≫ Fork.ι (coneOfIsSplitMono f) = Fork.ι s | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | dsimp | /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
-/
noncomputable def isSplitMonoEqualizes {X Y : C} (f : X ⟶ Y) [IsSplitMono f] :
IsLimit (coneOfIsSplitMono f) :=
Fork.IsLimit.mk' _ fun s =>
⟨s.ι ≫ retraction f, by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1204_0.eJEUq2AFfmN187w | /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
-/
noncomputable def isSplitMonoEqualizes {X Y : C} (f : X ⟶ Y) [IsSplitMono f] :
IsLimit (coneOfIsSplitMono f) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g : X✝ ⟶ Y✝
inst✝¹ : IsSplitMono f✝
X Y : C
f : X ⟶ Y
inst✝ : IsSplitMono f
s : Fork (𝟙 Y) (retraction f ≫ f)
⊢ (Fork.ι s ≫ retraction f) ≫ f = Fork.ι s | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rw [Category.assoc, ← s.condition] | /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
-/
noncomputable def isSplitMonoEqualizes {X Y : C} (f : X ⟶ Y) [IsSplitMono f] :
IsLimit (coneOfIsSplitMono f) :=
Fork.IsLimit.mk' _ fun s =>
⟨s.ι ≫ retraction f, by
dsimp
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1204_0.eJEUq2AFfmN187w | /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
-/
noncomputable def isSplitMonoEqualizes {X Y : C} (f : X ⟶ Y) [IsSplitMono f] :
IsLimit (coneOfIsSplitMono f) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g : X✝ ⟶ Y✝
inst✝¹ : IsSplitMono f✝
X Y : C
f : X ⟶ Y
inst✝ : IsSplitMono f
s : Fork (𝟙 Y) (retraction f ≫ f)
⊢ Fork.ι s ≫ 𝟙 Y = Fork.ι s | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | apply Category.comp_id | /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
-/
noncomputable def isSplitMonoEqualizes {X Y : C} (f : X ⟶ Y) [IsSplitMono f] :
IsLimit (coneOfIsSplitMono f) :=
Fork.IsLimit.mk' _ fun s =>
⟨s.ι ≫ retraction f, by
dsimp
rw [Category.assoc, ← s.condition]
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1204_0.eJEUq2AFfmN187w | /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
-/
noncomputable def isSplitMonoEqualizes {X Y : C} (f : X ⟶ Y) [IsSplitMono f] :
IsLimit (coneOfIsSplitMono f) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g : X✝ ⟶ Y✝
inst✝¹ : IsSplitMono f✝
X Y : C
f : X ⟶ Y
inst✝ : IsSplitMono f
s : Fork (𝟙 Y) (retraction f ≫ f)
m✝ :
((Functor.const WalkingParallelPair).obj s.pt).obj zero ⟶
((Functor.const WalkingParallelPair).obj (coneOfIsSplitMono f).pt).obj zero
hm : m✝ ≫ For... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp [← hm] | /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
-/
noncomputable def isSplitMonoEqualizes {X Y : C} (f : X ⟶ Y) [IsSplitMono f] :
IsLimit (coneOfIsSplitMono f) :=
Fork.IsLimit.mk' _ fun s =>
⟨s.ι ≫ retraction f, by
dsimp
rw [Category.assoc, ← s.condition]
apply Category.com... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1204_0.eJEUq2AFfmN187w | /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`.
-/
noncomputable def isSplitMonoEqualizes {X Y : C} (f : X ⟶ Y) [IsSplitMono f] :
IsLimit (coneOfIsSplitMono f) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Fork f g
i : IsLimit c
Z : C
h : Y ⟶ Z
hm : Mono h
⊢ Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp only [← Category.assoc] | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h := by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1228_0.eJEUq2AFfmN187w | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Fork f g
i : IsLimit c
Z : C
h : Y ⟶ Z
hm : Mono h
⊢ (Fork.ι c ≫ f) ≫ h = (Fork.ι c ≫ g) ≫ h | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | exact congrArg (· ≫ h) c.condition | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h := by
simp only [← Category.assoc]
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1228_0.eJEUq2AFfmN187w | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Fork f g
i : IsLimit c
Z : C
h : Y ⟶ Z
hm : Mono h
this : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h
⊢ Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp [this] | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h := by
simp only [← Category.assoc]
exact congrArg (· ≫ h) c.condition;
... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1228_0.eJEUq2AFfmN187w | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Fork f g
i : IsLimit c
Z : C
h : Y ⟶ Z
hm : Mono h
s : Fork (f ≫ h) (g ≫ h)
⊢ Fork.ι s ≫ f = Fork.ι s ≫ g | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | apply hm.right_cancellation | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h := by
simp only [← Category.assoc]
exact congrArg (· ≫ h) c.condition;
... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1228_0.eJEUq2AFfmN187w | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case a
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Fork f g
i : IsLimit c
Z : C
h : Y ⟶ Z
hm : Mono h
s : Fork (f ≫ h) (g ≫ h)
⊢ (Fork.ι s ≫ f) ≫ h = (Fork.ι s ≫ g) ≫ h | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp [s.condition] | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h := by
simp only [← Category.assoc]
exact congrArg (· ≫ h) c.condition;
... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1228_0.eJEUq2AFfmN187w | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Fork f g
i : IsLimit c
Z : C
h : Y ⟶ Z
hm✝ : Mono h
s : Fork (f ≫ h) (g ≫ h)
s' : Fork f g := Fork.ofι (Fork.ι s) (_ : Fork.ι s ≫ f = Fork.ι s ≫ g)
l : { l // l ≫ Fork.ι c = Fork.ι s' } := Fork.IsLimit.lift' i (Fork.ι s') (_ : Fork.ι s' ≫ f = Fork.ι s' ≫ g)
m... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | apply Fork.IsLimit.hom_ext i | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h := by
simp only [← Category.assoc]
exact congrArg (· ≫ h) c.condition;
... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1228_0.eJEUq2AFfmN187w | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Fork f g
i : IsLimit c
Z : C
h : Y ⟶ Z
hm✝ : Mono h
s : Fork (f ≫ h) (g ≫ h)
s' : Fork f g := Fork.ofι (Fork.ι s) (_ : Fork.ι s ≫ f = Fork.ι s ≫ g)
l : { l // l ≫ Fork.ι c = Fork.ι s' } := Fork.IsLimit.lift' i (Fork.ι s') (_ : Fork.ι s' ≫ f = Fork.ι s' ≫ g)
m... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rw [Fork.ι_ofι] at hm | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h := by
simp only [← Category.assoc]
exact congrArg (· ≫ h) c.condition;
... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1228_0.eJEUq2AFfmN187w | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Fork f g
i : IsLimit c
Z : C
h : Y ⟶ Z
hm✝ : Mono h
s : Fork (f ≫ h) (g ≫ h)
s' : Fork f g := Fork.ofι (Fork.ι s) (_ : Fork.ι s ≫ f = Fork.ι s ≫ g)
l : { l // l ≫ Fork.ι c = Fork.ι s' } := Fork.IsLimit.lift' i (Fork.ι s') (_ : Fork.ι s' ≫ f = Fork.ι s' ≫ g)
m... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rw [hm] | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h := by
simp only [← Category.assoc]
exact congrArg (· ≫ h) c.condition;
... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1228_0.eJEUq2AFfmN187w | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Fork f g
i : IsLimit c
Z : C
h : Y ⟶ Z
hm✝ : Mono h
s : Fork (f ≫ h) (g ≫ h)
s' : Fork f g := Fork.ofι (Fork.ι s) (_ : Fork.ι s ≫ f = Fork.ι s ≫ g)
l : { l // l ≫ Fork.ι c = Fork.ι s' } := Fork.IsLimit.lift' i (Fork.ι s') (_ : Fork.ι s' ≫ f = Fork.ι s' ≫ g)
m... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | exact l.2.symm | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h := by
simp only [← Category.assoc]
exact congrArg (· ≫ h) c.condition;
... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1228_0.eJEUq2AFfmN187w | /-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/
def isEqualizerCompMono {c : Fork f g} (i : IsLimit c) {Z : C} (h : Y ⟶ Z) [hm : Mono h] :
have : Fork.ι c ≫ f ≫ h = Fork.ι c ≫ g ≫ h | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X✝ Y : C
f✝ g : X✝ ⟶ Y
X : C
f : X ⟶ X
hf : f ≫ f = f
c : Fork (𝟙 X) f
i : IsLimit c
⊢ f ≫ 𝟙 X = f ≫ f | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp [hf] | /-- An equalizer of an idempotent morphism and the identity is split mono. -/
@[simps]
def splitMonoOfIdempotentOfIsLimitFork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Fork (𝟙 X) f}
(i : IsLimit c) : SplitMono c.ι where
retraction := i.lift (Fork.ofι f (by | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1250_0.eJEUq2AFfmN187w | /-- An equalizer of an idempotent morphism and the identity is split mono. -/
@[simps]
def splitMonoOfIdempotentOfIsLimitFork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Fork (𝟙 X) f}
(i : IsLimit c) : SplitMono c.ι where
retraction | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X✝ Y : C
f✝ g : X✝ ⟶ Y
X : C
f : X ⟶ X
hf : f ≫ f = f
c : Fork (𝟙 X) f
i : IsLimit c
⊢ Fork.ι c ≫ IsLimit.lift i (Fork.ofι f (_ : f ≫ 𝟙 X = f ≫ f)) =
𝟙 (((Functor.const WalkingParallelPair).obj c.pt).obj zero) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | letI := mono_of_isLimit_fork i | /-- An equalizer of an idempotent morphism and the identity is split mono. -/
@[simps]
def splitMonoOfIdempotentOfIsLimitFork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Fork (𝟙 X) f}
(i : IsLimit c) : SplitMono c.ι where
retraction := i.lift (Fork.ofι f (by simp [hf]))
id := by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1250_0.eJEUq2AFfmN187w | /-- An equalizer of an idempotent morphism and the identity is split mono. -/
@[simps]
def splitMonoOfIdempotentOfIsLimitFork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Fork (𝟙 X) f}
(i : IsLimit c) : SplitMono c.ι where
retraction | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X✝ Y : C
f✝ g : X✝ ⟶ Y
X : C
f : X ⟶ X
hf : f ≫ f = f
c : Fork (𝟙 X) f
i : IsLimit c
this : Mono (Fork.ι c) := mono_of_isLimit_fork i
⊢ Fork.ι c ≫ IsLimit.lift i (Fork.ofι f (_ : f ≫ 𝟙 X = f ≫ f)) =
𝟙 (((Functor.const WalkingParallelPair).obj c.pt).obj zero) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rw [← cancel_mono_id c.ι, Category.assoc, Fork.IsLimit.lift_ι, Fork.ι_ofι, ← c.condition] | /-- An equalizer of an idempotent morphism and the identity is split mono. -/
@[simps]
def splitMonoOfIdempotentOfIsLimitFork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Fork (𝟙 X) f}
(i : IsLimit c) : SplitMono c.ι where
retraction := i.lift (Fork.ofι f (by simp [hf]))
id := by
letI := mono_of_isLimit_fork ... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1250_0.eJEUq2AFfmN187w | /-- An equalizer of an idempotent morphism and the identity is split mono. -/
@[simps]
def splitMonoOfIdempotentOfIsLimitFork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Fork (𝟙 X) f}
(i : IsLimit c) : SplitMono c.ι where
retraction | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X✝ Y : C
f✝ g : X✝ ⟶ Y
X : C
f : X ⟶ X
hf : f ≫ f = f
c : Fork (𝟙 X) f
i : IsLimit c
this : Mono (Fork.ι c) := mono_of_isLimit_fork i
⊢ Fork.ι c ≫ 𝟙 X = Fork.ι c | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | exact Category.comp_id c.ι | /-- An equalizer of an idempotent morphism and the identity is split mono. -/
@[simps]
def splitMonoOfIdempotentOfIsLimitFork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Fork (𝟙 X) f}
(i : IsLimit c) : SplitMono c.ι where
retraction := i.lift (Fork.ofι f (by simp [hf]))
id := by
letI := mono_of_isLimit_fork ... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1250_0.eJEUq2AFfmN187w | /-- An equalizer of an idempotent morphism and the identity is split mono. -/
@[simps]
def splitMonoOfIdempotentOfIsLimitFork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Fork (𝟙 X) f}
(i : IsLimit c) : SplitMono c.ι where
retraction | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
inst✝ : IsSplitEpi f
⊢ 𝟙 X ≫ f = (f ≫ section_ f) ≫ f | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp | /-- A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`.
Here we build the cocone, and show in `isSplitEpiCoequalizes` that it is a colimit cocone.
-/
-- @[simps (config := { rhsMd := semireducible })] Porting note: no semireducible
@[simps!]
noncomputable def coconeOfIsSplitEpi : Cofork (𝟙 X) (f ≫ section_ f)... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1272_0.eJEUq2AFfmN187w | /-- A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`.
Here we build the cocone, and show in `isSplitEpiCoequalizes` that it is a colimit cocone.
-/
-- @[simps (config | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g : X✝ ⟶ Y✝
inst✝¹ : IsSplitEpi f✝
X Y : C
f : X ⟶ Y
inst✝ : IsSplitEpi f
s : Cofork (𝟙 X) (f ≫ section_ f)
⊢ Cofork.π (coconeOfIsSplitEpi f) ≫ section_ f ≫ Cofork.π s = Cofork.π s | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | dsimp | /-- A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`.
-/
noncomputable def isSplitEpiCoequalizes {X Y : C} (f : X ⟶ Y) [IsSplitEpi f] :
IsColimit (coconeOfIsSplitEpi f) :=
Cofork.IsColimit.mk' _ fun s =>
⟨section_ f ≫ s.π, by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1286_0.eJEUq2AFfmN187w | /-- A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`.
-/
noncomputable def isSplitEpiCoequalizes {X Y : C} (f : X ⟶ Y) [IsSplitEpi f] :
IsColimit (coconeOfIsSplitEpi f) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g : X✝ ⟶ Y✝
inst✝¹ : IsSplitEpi f✝
X Y : C
f : X ⟶ Y
inst✝ : IsSplitEpi f
s : Cofork (𝟙 X) (f ≫ section_ f)
⊢ f ≫ section_ f ≫ Cofork.π s = Cofork.π s | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rw [← Category.assoc, ← s.condition, Category.id_comp] | /-- A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`.
-/
noncomputable def isSplitEpiCoequalizes {X Y : C} (f : X ⟶ Y) [IsSplitEpi f] :
IsColimit (coconeOfIsSplitEpi f) :=
Cofork.IsColimit.mk' _ fun s =>
⟨section_ f ≫ s.π, by
dsimp
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1286_0.eJEUq2AFfmN187w | /-- A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`.
-/
noncomputable def isSplitEpiCoequalizes {X Y : C} (f : X ⟶ Y) [IsSplitEpi f] :
IsColimit (coconeOfIsSplitEpi f) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g : X✝ ⟶ Y✝
inst✝¹ : IsSplitEpi f✝
X Y : C
f : X ⟶ Y
inst✝ : IsSplitEpi f
s : Cofork (𝟙 X) (f ≫ section_ f)
m✝ :
((Functor.const WalkingParallelPair).obj (coconeOfIsSplitEpi f).pt).obj one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj one
hm : Cofork.π (c... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp [← hm] | /-- A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`.
-/
noncomputable def isSplitEpiCoequalizes {X Y : C} (f : X ⟶ Y) [IsSplitEpi f] :
IsColimit (coconeOfIsSplitEpi f) :=
Cofork.IsColimit.mk' _ fun s =>
⟨section_ f ≫ s.π, by
dsimp
rw [← Category.assoc, ← s.condition, Category.id_comp],... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1286_0.eJEUq2AFfmN187w | /-- A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`.
-/
noncomputable def isSplitEpiCoequalizes {X Y : C} (f : X ⟶ Y) [IsSplitEpi f] :
IsColimit (coconeOfIsSplitEpi f) | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Cofork f g
i : IsColimit c
W : C
h : W ⟶ X
hm : Epi h
⊢ (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp only [Category.assoc] | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c := by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1313_0.eJEUq2AFfmN187w | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Cofork f g
i : IsColimit c
W : C
h : W ⟶ X
hm : Epi h
⊢ h ≫ f ≫ Cofork.π c = h ≫ g ≫ Cofork.π c | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | exact congrArg (h ≫ ·) c.condition | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c := by
simp only [Category.assoc]
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1313_0.eJEUq2AFfmN187w | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Cofork f g
i : IsColimit c
W : C
h : W ⟶ X
hm : Epi h
s : Cofork (h ≫ f) (h ≫ g)
⊢ f ≫ Cofork.π s = g ≫ Cofork.π s | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | apply hm.left_cancellation | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c := by
simp only [Category.assoc]
exact congrArg (h ≫ ·) ... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1313_0.eJEUq2AFfmN187w | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
case a
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Cofork f g
i : IsColimit c
W : C
h : W ⟶ X
hm : Epi h
s : Cofork (h ≫ f) (h ≫ g)
⊢ h ≫ f ≫ Cofork.π s = h ≫ g ≫ Cofork.π s | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp_rw [← Category.assoc, s.condition] | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c := by
simp only [Category.assoc]
exact congrArg (h ≫ ·) ... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1313_0.eJEUq2AFfmN187w | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Cofork f g
i : IsColimit c
W : C
h : W ⟶ X
hm✝ : Epi h
s : Cofork (h ≫ f) (h ≫ g)
s' : Cofork f g := Cofork.ofπ (Cofork.π s) (_ : f ≫ Cofork.π s = g ≫ Cofork.π s)
l : { l // Cofork.π c ≫ l = Cofork.π s' } :=
Cofork.IsColimit.desc' i (Cofork.π s') (_ : f ≫ C... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | apply Cofork.IsColimit.hom_ext i | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c := by
simp only [Category.assoc]
exact congrArg (h ≫ ·) ... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1313_0.eJEUq2AFfmN187w | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Cofork f g
i : IsColimit c
W : C
h : W ⟶ X
hm✝ : Epi h
s : Cofork (h ≫ f) (h ≫ g)
s' : Cofork f g := Cofork.ofπ (Cofork.π s) (_ : f ≫ Cofork.π s = g ≫ Cofork.π s)
l : { l // Cofork.π c ≫ l = Cofork.π s' } :=
Cofork.IsColimit.desc' i (Cofork.π s') (_ : f ≫ C... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rw [Cofork.π_ofπ] at hm | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c := by
simp only [Category.assoc]
exact congrArg (h ≫ ·) ... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1313_0.eJEUq2AFfmN187w | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Cofork f g
i : IsColimit c
W : C
h : W ⟶ X
hm✝ : Epi h
s : Cofork (h ≫ f) (h ≫ g)
s' : Cofork f g := Cofork.ofπ (Cofork.π s) (_ : f ≫ Cofork.π s = g ≫ Cofork.π s)
l : { l // Cofork.π c ≫ l = Cofork.π s' } :=
Cofork.IsColimit.desc' i (Cofork.π s') (_ : f ≫ C... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rw [hm] | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c := by
simp only [Category.assoc]
exact congrArg (h ≫ ·) ... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1313_0.eJEUq2AFfmN187w | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
c : Cofork f g
i : IsColimit c
W : C
h : W ⟶ X
hm✝ : Epi h
s : Cofork (h ≫ f) (h ≫ g)
s' : Cofork f g := Cofork.ofπ (Cofork.π s) (_ : f ≫ Cofork.π s = g ≫ Cofork.π s)
l : { l // Cofork.π c ≫ l = Cofork.π s' } :=
Cofork.IsColimit.desc' i (Cofork.π s') (_ : f ≫ C... | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | exact l.2.symm | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c := by
simp only [Category.assoc]
exact congrArg (h ≫ ·) ... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1313_0.eJEUq2AFfmN187w | /-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is
a coequalizer. -/
def isCoequalizerEpiComp {c : Cofork f g} (i : IsColimit c) {W : C} (h : W ⟶ X) [hm : Epi h] :
have : (h ≫ f) ≫ Cofork.π c = (h ≫ g) ≫ Cofork.π c | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X✝ Y : C
f✝ g : X✝ ⟶ Y
X : C
f : X ⟶ X
hf : f ≫ f = f
c : Cofork (𝟙 X) f
i : IsColimit c
⊢ 𝟙 X ≫ f = f ≫ f | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | simp [hf] | /-- A coequalizer of an idempotent morphism and the identity is split epi. -/
@[simps]
def splitEpiOfIdempotentOfIsColimitCofork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Cofork (𝟙 X) f}
(i : IsColimit c) : SplitEpi c.π where
section_ := i.desc (Cofork.ofπ f (by | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1336_0.eJEUq2AFfmN187w | /-- A coequalizer of an idempotent morphism and the identity is split epi. -/
@[simps]
def splitEpiOfIdempotentOfIsColimitCofork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Cofork (𝟙 X) f}
(i : IsColimit c) : SplitEpi c.π where
section_ | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X✝ Y : C
f✝ g : X✝ ⟶ Y
X : C
f : X ⟶ X
hf : f ≫ f = f
c : Cofork (𝟙 X) f
i : IsColimit c
⊢ IsColimit.desc i (Cofork.ofπ f (_ : 𝟙 X ≫ f = f ≫ f)) ≫ Cofork.π c =
𝟙 (((Functor.const WalkingParallelPair).obj c.pt).obj one) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | letI := epi_of_isColimit_cofork i | /-- A coequalizer of an idempotent morphism and the identity is split epi. -/
@[simps]
def splitEpiOfIdempotentOfIsColimitCofork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Cofork (𝟙 X) f}
(i : IsColimit c) : SplitEpi c.π where
section_ := i.desc (Cofork.ofπ f (by simp [hf]))
id := by
| Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1336_0.eJEUq2AFfmN187w | /-- A coequalizer of an idempotent morphism and the identity is split epi. -/
@[simps]
def splitEpiOfIdempotentOfIsColimitCofork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Cofork (𝟙 X) f}
(i : IsColimit c) : SplitEpi c.π where
section_ | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X✝ Y : C
f✝ g : X✝ ⟶ Y
X : C
f : X ⟶ X
hf : f ≫ f = f
c : Cofork (𝟙 X) f
i : IsColimit c
this : Epi (Cofork.π c) := epi_of_isColimit_cofork i
⊢ IsColimit.desc i (Cofork.ofπ f (_ : 𝟙 X ≫ f = f ≫ f)) ≫ Cofork.π c =
𝟙 (((Functor.const WalkingParallelPair).obj c.pt).obj one) | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | rw [← cancel_epi_id c.π, ← Category.assoc, Cofork.IsColimit.π_desc, Cofork.π_ofπ, ←
c.condition] | /-- A coequalizer of an idempotent morphism and the identity is split epi. -/
@[simps]
def splitEpiOfIdempotentOfIsColimitCofork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Cofork (𝟙 X) f}
(i : IsColimit c) : SplitEpi c.π where
section_ := i.desc (Cofork.ofπ f (by simp [hf]))
id := by
letI := epi_of_isColimi... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1336_0.eJEUq2AFfmN187w | /-- A coequalizer of an idempotent morphism and the identity is split epi. -/
@[simps]
def splitEpiOfIdempotentOfIsColimitCofork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Cofork (𝟙 X) f}
(i : IsColimit c) : SplitEpi c.π where
section_ | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
C : Type u
inst✝ : Category.{v, u} C
X✝ Y : C
f✝ g : X✝ ⟶ Y
X : C
f : X ⟶ X
hf : f ≫ f = f
c : Cofork (𝟙 X) f
i : IsColimit c
this : Epi (Cofork.π c) := epi_of_isColimit_cofork i
⊢ 𝟙 X ≫ Cofork.π c = Cofork.π c | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | exact Category.id_comp _ | /-- A coequalizer of an idempotent morphism and the identity is split epi. -/
@[simps]
def splitEpiOfIdempotentOfIsColimitCofork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Cofork (𝟙 X) f}
(i : IsColimit c) : SplitEpi c.π where
section_ := i.desc (Cofork.ofπ f (by simp [hf]))
id := by
letI := epi_of_isColimi... | Mathlib.CategoryTheory.Limits.Shapes.Equalizers.1336_0.eJEUq2AFfmN187w | /-- A coequalizer of an idempotent morphism and the identity is split epi. -/
@[simps]
def splitEpiOfIdempotentOfIsColimitCofork {X : C} {f : X ⟶ X} (hf : f ≫ f = f) {c : Cofork (𝟙 X) f}
(i : IsColimit c) : SplitEpi c.π where
section_ | Mathlib_CategoryTheory_Limits_Shapes_Equalizers |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : Preorder α
⊢ Topology.IsUpperSet α | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | letI := upperSet α | instance [Preorder α] : @Topology.IsUpperSet α (upperSet α) _ := by
| Mathlib.Topology.Order.UpperLowerSetTopology.184_0.8VIerEucs6khyhO | instance [Preorder α] : @Topology.IsUpperSet α (upperSet α) _ | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : Preorder α
this : TopologicalSpace α := upperSet α
⊢ Topology.IsUpperSet α | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | exact ⟨rfl⟩ | instance [Preorder α] : @Topology.IsUpperSet α (upperSet α) _ := by
letI := upperSet α
| Mathlib.Topology.Order.UpperLowerSetTopology.184_0.8VIerEucs6khyhO | instance [Preorder α] : @Topology.IsUpperSet α (upperSet α) _ | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : Preorder α
⊢ Topology.IsLowerSet α | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | letI := lowerSet α | instance [Preorder α] : @Topology.IsLowerSet α (lowerSet α) _ := by
| Mathlib.Topology.Order.UpperLowerSetTopology.199_0.8VIerEucs6khyhO | instance [Preorder α] : @Topology.IsLowerSet α (lowerSet α) _ | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : Preorder α
this : TopologicalSpace α := lowerSet α
⊢ Topology.IsLowerSet α | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | exact ⟨rfl⟩ | instance [Preorder α] : @Topology.IsLowerSet α (lowerSet α) _ := by
letI := lowerSet α
| Mathlib.Topology.Order.UpperLowerSetTopology.199_0.8VIerEucs6khyhO | instance [Preorder α] : @Topology.IsLowerSet α (lowerSet α) _ | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Topology.IsUpperSet α
s : Set α
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
⊢ instTopologicalSpaceOrderDual = lowerSet αᵒᵈ | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | ext | instance _root_.OrderDual.instIsLowerSet [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] :
Topology.IsLowerSet αᵒᵈ where
topology_eq_lowerSetTopology := by | Mathlib.Topology.Order.UpperLowerSetTopology.214_0.8VIerEucs6khyhO | instance _root_.OrderDual.instIsLowerSet [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] :
Topology.IsLowerSet αᵒᵈ where
topology_eq_lowerSetTopology | Mathlib_Topology_Order_UpperLowerSetTopology |
case a.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Topology.IsUpperSet α
s : Set α
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
x✝ : Set αᵒᵈ
⊢ IsOpen x✝ ↔ IsOpen x✝ | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [IsUpperSet.topology_eq α] | instance _root_.OrderDual.instIsLowerSet [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] :
Topology.IsLowerSet αᵒᵈ where
topology_eq_lowerSetTopology := by ext; | Mathlib.Topology.Order.UpperLowerSetTopology.214_0.8VIerEucs6khyhO | instance _root_.OrderDual.instIsLowerSet [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] :
Topology.IsLowerSet αᵒᵈ where
topology_eq_lowerSetTopology | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s : Set α
⊢ WithUpperSet.instTopologicalSpaceWithUpperSet = induced (⇑WithUpperSet.ofUpperSet) inst✝¹ | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | erw [topology_eq α, induced_id] | /-- If `α` is equipped with the upper set topology, then it is homeomorphic to
`WithUpperSet α`. -/
def WithUpperSetHomeomorph : WithUpperSet α ≃ₜ α :=
WithUpperSet.ofUpperSet.toHomeomorphOfInducing ⟨by | Mathlib.Topology.Order.UpperLowerSetTopology.218_0.8VIerEucs6khyhO | /-- If `α` is equipped with the upper set topology, then it is homeomorphic to
`WithUpperSet α`. -/
def WithUpperSetHomeomorph : WithUpperSet α ≃ₜ α | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s : Set α
⊢ WithUpperSet.instTopologicalSpaceWithUpperSet = upperSet α | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rfl | /-- If `α` is equipped with the upper set topology, then it is homeomorphic to
`WithUpperSet α`. -/
def WithUpperSetHomeomorph : WithUpperSet α ≃ₜ α :=
WithUpperSet.ofUpperSet.toHomeomorphOfInducing ⟨by erw [topology_eq α, induced_id]; | Mathlib.Topology.Order.UpperLowerSetTopology.218_0.8VIerEucs6khyhO | /-- If `α` is equipped with the upper set topology, then it is homeomorphic to
`WithUpperSet α`. -/
def WithUpperSetHomeomorph : WithUpperSet α ≃ₜ α | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s : Set α
⊢ IsOpen s ↔ IsUpperSet s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [topology_eq α] | lemma isOpen_iff_isUpperSet : IsOpen s ↔ IsUpperSet s := by
| Mathlib.Topology.Order.UpperLowerSetTopology.223_0.8VIerEucs6khyhO | lemma isOpen_iff_isUpperSet : IsOpen s ↔ IsUpperSet s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s : Set α
⊢ IsOpen s ↔ IsUpperSet s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rfl | lemma isOpen_iff_isUpperSet : IsOpen s ↔ IsUpperSet s := by
rw [topology_eq α]
| Mathlib.Topology.Order.UpperLowerSetTopology.223_0.8VIerEucs6khyhO | lemma isOpen_iff_isUpperSet : IsOpen s ↔ IsUpperSet s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s : Set α
S : Set (Set α)
⊢ (∀ s ∈ S, IsOpen s) → IsOpen (⋂₀ S) | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | simpa only [isOpen_iff_isUpperSet] using isUpperSet_sInter (α := α) | instance toAlexandrovDiscrete : AlexandrovDiscrete α where
isOpen_sInter S := by | Mathlib.Topology.Order.UpperLowerSetTopology.227_0.8VIerEucs6khyhO | instance toAlexandrovDiscrete : AlexandrovDiscrete α where
isOpen_sInter S | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s : Set α
⊢ IsClosed s ↔ IsLowerSet s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [← isOpen_compl_iff, isOpen_iff_isUpperSet,
isLowerSet_compl.symm, compl_compl] | lemma isClosed_iff_isLower : IsClosed s ↔ IsLowerSet s := by
| Mathlib.Topology.Order.UpperLowerSetTopology.231_0.8VIerEucs6khyhO | lemma isClosed_iff_isLower : IsClosed s ↔ IsLowerSet s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s✝ s : Set α
⊢ closure s = ↑(lowerClosure s) | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [subset_antisymm_iff] | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s := by
| Mathlib.Topology.Order.UpperLowerSetTopology.235_0.8VIerEucs6khyhO | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s✝ s : Set α
⊢ closure s ⊆ ↑(lowerClosure s) ∧ ↑(lowerClosure s) ⊆ closure s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | refine ⟨?_, lowerClosure_min subset_closure (isClosed_iff_isLower.1 isClosed_closure)⟩ | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s := by
rw [subset_antisymm_iff]
| Mathlib.Topology.Order.UpperLowerSetTopology.235_0.8VIerEucs6khyhO | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s✝ s : Set α
⊢ closure s ⊆ ↑(lowerClosure s) | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | apply closure_minimal subset_lowerClosure _ | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s := by
rw [subset_antisymm_iff]
refine ⟨?_, lowerClosure_min subset_closure (isClosed_iff_isLower.1 isClosed_closure)⟩
· | Mathlib.Topology.Order.UpperLowerSetTopology.235_0.8VIerEucs6khyhO | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s✝ s : Set α
⊢ IsClosed ↑(lowerClosure s) | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [isClosed_iff_isLower] | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s := by
rw [subset_antisymm_iff]
refine ⟨?_, lowerClosure_min subset_closure (isClosed_iff_isLower.1 isClosed_closure)⟩
· apply closure_minimal subset_lowerClosure _
| Mathlib.Topology.Order.UpperLowerSetTopology.235_0.8VIerEucs6khyhO | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s✝ s : Set α
⊢ IsLowerSet ↑(lowerClosure s) | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | exact LowerSet.lower (lowerClosure s) | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s := by
rw [subset_antisymm_iff]
refine ⟨?_, lowerClosure_min subset_closure (isClosed_iff_isLower.1 isClosed_closure)⟩
· apply closure_minimal subset_lowerClosure _
rw [isClosed_iff_isLower]
| Mathlib.Topology.Order.UpperLowerSetTopology.235_0.8VIerEucs6khyhO | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s : Set α
a : α
⊢ closure {a} = Iic a | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [closure_eq_lowerClosure, lowerClosure_singleton] | /--
The closure of a singleton `{a}` in the upper set topology is the right-closed left-infinite
interval (-∞,a].
-/
@[simp] lemma closure_singleton {a : α} : closure {a} = Iic a := by
| Mathlib.Topology.Order.UpperLowerSetTopology.242_0.8VIerEucs6khyhO | /--
The closure of a singleton `{a}` in the upper set topology is the right-closed left-infinite
interval (-∞,a].
-/
@[simp] lemma closure_singleton {a : α} : closure {a} = Iic a | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s : Set α
a : α
⊢ ↑(LowerSet.Iic a) = Iic a | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rfl | /--
The closure of a singleton `{a}` in the upper set topology is the right-closed left-infinite
interval (-∞,a].
-/
@[simp] lemma closure_singleton {a : α} : closure {a} = Iic a := by
rw [closure_eq_lowerClosure, lowerClosure_singleton]
| Mathlib.Topology.Order.UpperLowerSetTopology.242_0.8VIerEucs6khyhO | /--
The closure of a singleton `{a}` in the upper set topology is the right-closed left-infinite
interval (-∞,a].
-/
@[simp] lemma closure_singleton {a : α} : closure {a} = Iic a | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
⊢ Monotone f ↔ Continuous f | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | constructor | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
| Mathlib.Topology.Order.UpperLowerSetTopology.258_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
⊢ Monotone f → Continuous f | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | intro hf | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
constructor
· | Mathlib.Topology.Order.UpperLowerSetTopology.258_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Monotone f
⊢ Continuous f | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | simp_rw [continuous_def, isOpen_iff_isUpperSet] | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
constructor
· intro hf
| Mathlib.Topology.Order.UpperLowerSetTopology.258_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Monotone f
⊢ ∀ (s : Set β), IsUpperSet s → IsUpperSet (f ⁻¹' s) | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | exact fun _ hs ↦ IsUpperSet.preimage hs hf | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
constructor
· intro hf
simp_rw [continuous_def, isOpen_iff_isUpperSet]
| Mathlib.Topology.Order.UpperLowerSetTopology.258_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
⊢ Continuous f → Monotone f | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | intro hf a b hab | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
constructor
· intro hf
simp_rw [continuous_def, isOpen_iff_isUpperSet]
exact fun _ hs ↦ IsUpperSet.preimage hs hf
· | Mathlib.Topology.Order.UpperLowerSetTopology.258_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Continuous f
a b : α
hab : a ≤ b
⊢ f a ≤ f b | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [← mem_Iic, ← closure_singleton] at hab ⊢ | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
constructor
· intro hf
simp_rw [continuous_def, isOpen_iff_isUpperSet]
exact fun _ hs ↦ IsUpperSet.preimage hs hf
· intro hf a... | Mathlib.Topology.Order.UpperLowerSetTopology.258_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Continuous f
a b : α
hab : a ∈ closure {b}
⊢ f a ∈ closure {f b} | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | apply (Continuous.closure_preimage_subset hf {f b}) | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
constructor
· intro hf
simp_rw [continuous_def, isOpen_iff_isUpperSet]
exact fun _ hs ↦ IsUpperSet.preimage hs hf
· intro hf a... | Mathlib.Topology.Order.UpperLowerSetTopology.258_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
case mpr.a
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Continuous f
a b : α
hab : a ∈ closure {b}
⊢ a ∈ closure (f ⁻¹' {f b}) | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | apply mem_of_mem_of_subset hab | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
constructor
· intro hf
simp_rw [continuous_def, isOpen_iff_isUpperSet]
exact fun _ hs ↦ IsUpperSet.preimage hs hf
· intro hf a... | Mathlib.Topology.Order.UpperLowerSetTopology.258_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
case mpr.a
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Continuous f
a b : α
hab : a ∈ closure {b}
⊢ closure {b} ⊆ closure (f ⁻¹' {f b}) | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | apply closure_mono | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
constructor
· intro hf
simp_rw [continuous_def, isOpen_iff_isUpperSet]
exact fun _ hs ↦ IsUpperSet.preimage hs hf
· intro hf a... | Mathlib.Topology.Order.UpperLowerSetTopology.258_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
case mpr.a.h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Continuous f
a b : α
hab : a ∈ closure {b}
⊢ {b} ⊆ f ⁻¹' {f b} | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [singleton_subset_iff, mem_preimage, mem_singleton_iff] | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by
constructor
· intro hf
simp_rw [continuous_def, isOpen_iff_isUpperSet]
exact fun _ hs ↦ IsUpperSet.preimage hs hf
· intro hf a... | Mathlib.Topology.Order.UpperLowerSetTopology.258_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : IsUpper β
f : α → β
hf : Monotone f
⊢ Continuous f | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | simp_rw [continuous_def, isOpen_iff_isUpperSet] | lemma monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f := by
| Mathlib.Topology.Order.UpperLowerSetTopology.271_0.8VIerEucs6khyhO | lemma monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : IsUpper β
f : α → β
hf : Monotone f
⊢ ∀ (s : Set β), IsOpen s → IsUpperSet (f ⁻¹' s) | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | intro s hs | lemma monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f := by
simp_rw [continuous_def, isOpen_iff_isUpperSet]
| Mathlib.Topology.Order.UpperLowerSetTopology.271_0.8VIerEucs6khyhO | lemma monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : IsUpper β
f : α → β
hf : Monotone f
s : Set β
hs : IsOpen s
⊢ IsUpperSet (f ⁻¹' s) | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | exact (IsUpper.isUpperSet_of_isOpen hs).preimage hf | lemma monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f := by
simp_rw [continuous_def, isOpen_iff_isUpperSet]
intro s hs
| Mathlib.Topology.Order.UpperLowerSetTopology.271_0.8VIerEucs6khyhO | lemma monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
t₁ t₂ : TopologicalSpace α
inst✝¹ : Topology.IsUpperSet α
inst✝ : IsUpper α
s : Set α
hs : IsOpen s
⊢ IsOpen s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [@isOpen_iff_isUpperSet α _ t₁] | lemma upperSet_le_upper {t₁ t₂ : TopologicalSpace α} [@Topology.IsUpperSet α t₁ _]
[@Topology.IsUpper α t₂ _] : t₁ ≤ t₂ := fun s hs => by
| Mathlib.Topology.Order.UpperLowerSetTopology.277_0.8VIerEucs6khyhO | lemma upperSet_le_upper {t₁ t₂ : TopologicalSpace α} [@Topology.IsUpperSet α t₁ _]
[@Topology.IsUpper α t₂ _] : t₁ ≤ t₂ | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
t₁ t₂ : TopologicalSpace α
inst✝¹ : Topology.IsUpperSet α
inst✝ : IsUpper α
s : Set α
hs : IsOpen s
⊢ IsUpperSet s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | exact IsUpper.isUpperSet_of_isOpen hs | lemma upperSet_le_upper {t₁ t₂ : TopologicalSpace α} [@Topology.IsUpperSet α t₁ _]
[@Topology.IsUpper α t₂ _] : t₁ ≤ t₂ := fun s hs => by
rw [@isOpen_iff_isUpperSet α _ t₁]
| Mathlib.Topology.Order.UpperLowerSetTopology.277_0.8VIerEucs6khyhO | lemma upperSet_le_upper {t₁ t₂ : TopologicalSpace α} [@Topology.IsUpperSet α t₁ _]
[@Topology.IsUpper α t₂ _] : t₁ ≤ t₂ | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Topology.IsLowerSet α
s : Set α
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsLowerSet α
⊢ instTopologicalSpaceOrderDual = upperSet αᵒᵈ | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | ext | instance _root_.OrderDual.instIsUpperSet [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] :
Topology.IsUpperSet αᵒᵈ where
topology_eq_upperSetTopology := by | Mathlib.Topology.Order.UpperLowerSetTopology.297_0.8VIerEucs6khyhO | instance _root_.OrderDual.instIsUpperSet [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] :
Topology.IsUpperSet αᵒᵈ where
topology_eq_upperSetTopology | Mathlib_Topology_Order_UpperLowerSetTopology |
case a.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Topology.IsLowerSet α
s : Set α
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsLowerSet α
x✝ : Set αᵒᵈ
⊢ IsOpen x✝ ↔ IsOpen x✝ | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [IsLowerSet.topology_eq α] | instance _root_.OrderDual.instIsUpperSet [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] :
Topology.IsUpperSet αᵒᵈ where
topology_eq_upperSetTopology := by ext; | Mathlib.Topology.Order.UpperLowerSetTopology.297_0.8VIerEucs6khyhO | instance _root_.OrderDual.instIsUpperSet [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] :
Topology.IsUpperSet αᵒᵈ where
topology_eq_upperSetTopology | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsLowerSet α
s : Set α
⊢ WithLowerSet.instTopologicalSpaceWithLowerSet = induced (⇑WithLowerSet.ofLowerSet) inst✝¹ | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | erw [topology_eq α, induced_id] | /-- If `α` is equipped with the lower set topology, then it is homeomorphic to `WithLowerSet α`. -/
def WithLowerSetHomeomorph : WithLowerSet α ≃ₜ α :=
WithLowerSet.ofLowerSet.toHomeomorphOfInducing ⟨by | Mathlib.Topology.Order.UpperLowerSetTopology.301_0.8VIerEucs6khyhO | /-- If `α` is equipped with the lower set topology, then it is homeomorphic to `WithLowerSet α`. -/
def WithLowerSetHomeomorph : WithLowerSet α ≃ₜ α | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsLowerSet α
s : Set α
⊢ WithLowerSet.instTopologicalSpaceWithLowerSet = lowerSet α | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rfl | /-- If `α` is equipped with the lower set topology, then it is homeomorphic to `WithLowerSet α`. -/
def WithLowerSetHomeomorph : WithLowerSet α ≃ₜ α :=
WithLowerSet.ofLowerSet.toHomeomorphOfInducing ⟨by erw [topology_eq α, induced_id]; | Mathlib.Topology.Order.UpperLowerSetTopology.301_0.8VIerEucs6khyhO | /-- If `α` is equipped with the lower set topology, then it is homeomorphic to `WithLowerSet α`. -/
def WithLowerSetHomeomorph : WithLowerSet α ≃ₜ α | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsLowerSet α
s : Set α
⊢ IsOpen s ↔ IsLowerSet s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [topology_eq α] | lemma isOpen_iff_isLowerSet : IsOpen s ↔ IsLowerSet s := by | Mathlib.Topology.Order.UpperLowerSetTopology.305_0.8VIerEucs6khyhO | lemma isOpen_iff_isLowerSet : IsOpen s ↔ IsLowerSet s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsLowerSet α
s : Set α
⊢ IsOpen s ↔ IsLowerSet s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rfl | lemma isOpen_iff_isLowerSet : IsOpen s ↔ IsLowerSet s := by rw [topology_eq α]; | Mathlib.Topology.Order.UpperLowerSetTopology.305_0.8VIerEucs6khyhO | lemma isOpen_iff_isLowerSet : IsOpen s ↔ IsLowerSet s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsLowerSet α
s : Set α
⊢ IsClosed s ↔ IsUpperSet s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [← isOpen_compl_iff, isOpen_iff_isLowerSet, isUpperSet_compl.symm, compl_compl] | lemma isClosed_iff_isUpper : IsClosed s ↔ IsUpperSet s := by
| Mathlib.Topology.Order.UpperLowerSetTopology.309_0.8VIerEucs6khyhO | lemma isClosed_iff_isUpper : IsClosed s ↔ IsUpperSet s | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsLowerSet α
s : Set α
a : α
⊢ closure {a} = Ici a | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [closure_eq_upperClosure, upperClosure_singleton] | /--
The closure of a singleton `{a}` in the lower set topology is the right-closed left-infinite
interval (-∞,a].
-/
@[simp] lemma closure_singleton {a : α} : closure {a} = Ici a := by
| Mathlib.Topology.Order.UpperLowerSetTopology.315_0.8VIerEucs6khyhO | /--
The closure of a singleton `{a}` in the lower set topology is the right-closed left-infinite
interval (-∞,a].
-/
@[simp] lemma closure_singleton {a : α} : closure {a} = Ici a | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsLowerSet α
s : Set α
a : α
⊢ ↑(UpperSet.Ici a) = Ici a | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rfl | /--
The closure of a singleton `{a}` in the lower set topology is the right-closed left-infinite
interval (-∞,a].
-/
@[simp] lemma closure_singleton {a : α} : closure {a} = Ici a := by
rw [closure_eq_upperClosure, upperClosure_singleton]
| Mathlib.Topology.Order.UpperLowerSetTopology.315_0.8VIerEucs6khyhO | /--
The closure of a singleton `{a}` in the lower set topology is the right-closed left-infinite
interval (-∞,a].
-/
@[simp] lemma closure_singleton {a : α} : closure {a} = Ici a | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsLowerSet α
inst✝ : Topology.IsLowerSet β
f : α → β
⊢ Monotone f ↔ Continuous f | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [← monotone_dual_iff] | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsLowerSet α] [Topology.IsLowerSet β] {f : α → β} : Monotone f ↔ Continuous f := by
| Mathlib.Topology.Order.UpperLowerSetTopology.332_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsLowerSet α] [Topology.IsLowerSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsLowerSet α
inst✝ : Topology.IsLowerSet β
f : α → β
⊢ Monotone (⇑toDual ∘ f ∘ ⇑ofDual) ↔ Continuous f | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | exact IsUpperSet.monotone_iff_continuous (α := αᵒᵈ) (β := βᵒᵈ)
(f := (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ)) | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsLowerSet α] [Topology.IsLowerSet β] {f : α → β} : Monotone f ↔ Continuous f := by
rw [← monotone_dual_iff]
| Mathlib.Topology.Order.UpperLowerSetTopology.332_0.8VIerEucs6khyhO | protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β]
[Topology.IsLowerSet α] [Topology.IsLowerSet β] {f : α → β} : Monotone f ↔ Continuous f | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
t₁ t₂ : TopologicalSpace α
inst✝¹ : Topology.IsLowerSet α
inst✝ : IsLower α
s : Set α
hs : IsOpen s
⊢ IsOpen s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | rw [@isOpen_iff_isLowerSet α _ t₁] | lemma lowerSet_le_lower {t₁ t₂ : TopologicalSpace α} [@Topology.IsLowerSet α t₁ _]
[@IsLower α t₂ _] : t₁ ≤ t₂ := fun s hs => by
| Mathlib.Topology.Order.UpperLowerSetTopology.342_0.8VIerEucs6khyhO | lemma lowerSet_le_lower {t₁ t₂ : TopologicalSpace α} [@Topology.IsLowerSet α t₁ _]
[@IsLower α t₂ _] : t₁ ≤ t₂ | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
t₁ t₂ : TopologicalSpace α
inst✝¹ : Topology.IsLowerSet α
inst✝ : IsLower α
s : Set α
hs : IsOpen s
⊢ IsLowerSet s | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | exact IsLower.isLowerSet_of_isOpen hs | lemma lowerSet_le_lower {t₁ t₂ : TopologicalSpace α} [@Topology.IsLowerSet α t₁ _]
[@IsLower α t₂ _] : t₁ ≤ t₂ := fun s hs => by
rw [@isOpen_iff_isLowerSet α _ t₁]
| Mathlib.Topology.Order.UpperLowerSetTopology.342_0.8VIerEucs6khyhO | lemma lowerSet_le_lower {t₁ t₂ : TopologicalSpace α} [@Topology.IsLowerSet α t₁ _]
[@IsLower α t₂ _] : t₁ ≤ t₂ | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | constructor | lemma isUpperSet_orderDual [Preorder α] [TopologicalSpace α] :
Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α := by
| Mathlib.Topology.Order.UpperLowerSetTopology.351_0.8VIerEucs6khyhO | lemma isUpperSet_orderDual [Preorder α] [TopologicalSpace α] :
Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α | Mathlib_Topology_Order_UpperLowerSetTopology |
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ Topology.IsUpperSet αᵒᵈ → Topology.IsLowerSet α | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | apply OrderDual.instIsLowerSet | lemma isUpperSet_orderDual [Preorder α] [TopologicalSpace α] :
Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α := by
constructor
· | Mathlib.Topology.Order.UpperLowerSetTopology.351_0.8VIerEucs6khyhO | lemma isUpperSet_orderDual [Preorder α] [TopologicalSpace α] :
Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α | Mathlib_Topology_Order_UpperLowerSetTopology |
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ Topology.IsLowerSet α → Topology.IsUpperSet αᵒᵈ | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | apply OrderDual.instIsUpperSet | lemma isUpperSet_orderDual [Preorder α] [TopologicalSpace α] :
Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α := by
constructor
· apply OrderDual.instIsLowerSet
· | Mathlib.Topology.Order.UpperLowerSetTopology.351_0.8VIerEucs6khyhO | lemma isUpperSet_orderDual [Preorder α] [TopologicalSpace α] :
Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : Preorder γ
a b : α
⊢ toUpperSet a ⤳ toUpperSet b ↔ b ≤ a | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | simp_rw [specializes_iff_closure_subset, IsUpperSet.closure_singleton, Iic_subset_Iic,
toUpperSet_le_iff] | @[simp] lemma toUpperSet_specializes_toUpperSet {a b : α} :
toUpperSet a ⤳ toUpperSet b ↔ b ≤ a := by
| Mathlib.Topology.Order.UpperLowerSetTopology.372_0.8VIerEucs6khyhO | @[simp] lemma toUpperSet_specializes_toUpperSet {a b : α} :
toUpperSet a ⤳ toUpperSet b ↔ b ≤ a | Mathlib_Topology_Order_UpperLowerSetTopology |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : Preorder γ
a b : α
⊢ toLowerSet a ⤳ toLowerSet b ↔ a ≤ b | /-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and ... | simp_rw [specializes_iff_closure_subset, IsLowerSet.closure_singleton, Ici_subset_Ici,
toLowerSet_le_iff] | @[simp] lemma toLowerSet_specializes_toLowerSet {a b : α} :
toLowerSet a ⤳ toLowerSet b ↔ a ≤ b := by
| Mathlib.Topology.Order.UpperLowerSetTopology.400_0.8VIerEucs6khyhO | @[simp] lemma toLowerSet_specializes_toLowerSet {a b : α} :
toLowerSet a ⤳ toLowerSet b ↔ a ≤ b | Mathlib_Topology_Order_UpperLowerSetTopology |
U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
e : u ⟶ v
⊢ (u ⟶ v) = (u' ⟶ v') | /-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | {rw [hu, hv]} | theorem Hom.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
e.cast hu hv = _root_.cast (by | Mathlib.Combinatorics.Quiver.Cast.38_0.D9XIi49CIzM7YYf | theorem Hom.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
e.cast hu hv = _root_.cast (by {rw [hu, hv]}) e | Mathlib_Combinatorics_Quiver_Cast |
U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
e : u ⟶ v
⊢ (u ⟶ v) = (u' ⟶ v') | /-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | rw [hu, hv] | theorem Hom.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
e.cast hu hv = _root_.cast (by { | Mathlib.Combinatorics.Quiver.Cast.38_0.D9XIi49CIzM7YYf | theorem Hom.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
e.cast hu hv = _root_.cast (by {rw [hu, hv]}) e | Mathlib_Combinatorics_Quiver_Cast |
U : Type u_1
inst✝ : Quiver U
u v u' v' : U
hu : u = u'
hv : v = v'
e : u ⟶ v
⊢ cast hu hv e = _root_.cast (_ : (u ⟶ v) = (u' ⟶ v')) e | /-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | subst_vars | theorem Hom.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
e.cast hu hv = _root_.cast (by {rw [hu, hv]}) e := by
| Mathlib.Combinatorics.Quiver.Cast.38_0.D9XIi49CIzM7YYf | theorem Hom.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
e.cast hu hv = _root_.cast (by {rw [hu, hv]}) e | Mathlib_Combinatorics_Quiver_Cast |
U : Type u_1
inst✝ : Quiver U
u' v' : U
e : u' ⟶ v'
⊢ cast (_ : u' = u') (_ : v' = v') e = _root_.cast (_ : (u' ⟶ v') = (u' ⟶ v')) e | /-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | rfl | theorem Hom.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
e.cast hu hv = _root_.cast (by {rw [hu, hv]}) e := by
subst_vars
| Mathlib.Combinatorics.Quiver.Cast.38_0.D9XIi49CIzM7YYf | theorem Hom.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
e.cast hu hv = _root_.cast (by {rw [hu, hv]}) e | Mathlib_Combinatorics_Quiver_Cast |
U : Type u_1
inst✝ : Quiver U
u v u' v' u'' v'' : U
e : u ⟶ v
hu : u = u'
hv : v = v'
hu' : u' = u''
hv' : v' = v''
⊢ cast hu' hv' (cast hu hv e) = cast (_ : u = u'') (_ : v = v'') e | /-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | subst_vars | @[simp]
theorem Hom.cast_cast {u v u' v' u'' v'' : U} (e : u ⟶ v) (hu : u = u') (hv : v = v')
(hu' : u' = u'') (hv' : v' = v'') :
(e.cast hu hv).cast hu' hv' = e.cast (hu.trans hu') (hv.trans hv') := by
| Mathlib.Combinatorics.Quiver.Cast.49_0.D9XIi49CIzM7YYf | @[simp]
theorem Hom.cast_cast {u v u' v' u'' v'' : U} (e : u ⟶ v) (hu : u = u') (hv : v = v')
(hu' : u' = u'') (hv' : v' = v'') :
(e.cast hu hv).cast hu' hv' = e.cast (hu.trans hu') (hv.trans hv') | Mathlib_Combinatorics_Quiver_Cast |
U : Type u_1
inst✝ : Quiver U
u'' v'' : U
e : u'' ⟶ v''
⊢ cast (_ : u'' = u'') (_ : v'' = v'') (cast (_ : u'' = u'') (_ : v'' = v'') e) = cast (_ : u'' = u'') (_ : v'' = v'') e | /-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | rfl | @[simp]
theorem Hom.cast_cast {u v u' v' u'' v'' : U} (e : u ⟶ v) (hu : u = u') (hv : v = v')
(hu' : u' = u'') (hv' : v' = v'') :
(e.cast hu hv).cast hu' hv' = e.cast (hu.trans hu') (hv.trans hv') := by
subst_vars
| Mathlib.Combinatorics.Quiver.Cast.49_0.D9XIi49CIzM7YYf | @[simp]
theorem Hom.cast_cast {u v u' v' u'' v'' : U} (e : u ⟶ v) (hu : u = u') (hv : v = v')
(hu' : u' = u'') (hv' : v' = v'') :
(e.cast hu hv).cast hu' hv' = e.cast (hu.trans hu') (hv.trans hv') | Mathlib_Combinatorics_Quiver_Cast |
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