state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case w.w
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
⊢ ((fun j_1 k => if h : j = k then eqToHom (_ : X M j = X M k) else 0) ≫ fun j k =>
if h : j = i then eqToHom (_ : X M j = X M i) else 0) =
if h : j = i then eqToHom (_ : (embedding C).obj (X M j) = (embedding C).obj ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | ext ⟨⟩ ⟨⟩ | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom := biproduct.lift fun i j k => if h : j = i then eqToHom (congr_arg M.X h) else 0
inv := biproduct.desc fun i j k => if h : i = k then ... | Mathlib.CategoryTheory.Preadditive.Mat.332_0.xG9GKY7NTklnF73 | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom | Mathlib_CategoryTheory_Preadditive_Mat |
case w.w.H.unit.unit
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
⊢ ((fun j_1 k => if h : j = k then eqToHom (_ : X M j = X M k) else 0) ≫ fun j k =>
if h : j = i then eqToHom (_ : X M j = X M i) else 0)
PUnit.unit PUnit.unit =
dite (j = i) (fun h => eqToHom (_ :... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp only [embedding, comp_apply, comp_dite, dite_comp, comp_zero, zero_comp,
Finset.sum_dite_eq', Finset.mem_univ, ite_true, eqToHom_refl, Category.comp_id] | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom := biproduct.lift fun i j k => if h : j = i then eqToHom (congr_arg M.X h) else 0
inv := biproduct.desc fun i j k => if h : i = k then ... | Mathlib.CategoryTheory.Preadditive.Mat.332_0.xG9GKY7NTklnF73 | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom | Mathlib_CategoryTheory_Preadditive_Mat |
case w.w.H.unit.unit
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
⊢ (if h : j = i then eqToHom (_ : X M j = X M i) else 0) =
dite (j = i) (fun h => eqToHom (_ : (embedding C).obj (X M j) = (embedding C).obj (X M i))) (fun h => 0) PUnit.unit
PUnit.unit | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | split_ifs with h | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom := biproduct.lift fun i j k => if h : j = i then eqToHom (congr_arg M.X h) else 0
inv := biproduct.desc fun i j k => if h : i = k then ... | Mathlib.CategoryTheory.Preadditive.Mat.332_0.xG9GKY7NTklnF73 | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
h : j = i
⊢ eqToHom (_ : X M j = X M i) =
eqToHom (_ : (embedding C).obj (X M j) = (embedding C).obj (X M i)) PUnit.unit PUnit.unit | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | subst h | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom := biproduct.lift fun i j k => if h : j = i then eqToHom (congr_arg M.X h) else 0
inv := biproduct.desc fun i j k => if h : i = k then ... | Mathlib.CategoryTheory.Preadditive.Mat.332_0.xG9GKY7NTklnF73 | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
j : M.ι
⊢ eqToHom (_ : X M j = X M j) =
eqToHom (_ : (embedding C).obj (X M j) = (embedding C).obj (X M j)) PUnit.unit PUnit.unit | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom := biproduct.lift fun i j k => if h : j = i then eqToHom (congr_arg M.X h) else 0
inv := biproduct.desc fun i j k => if h : i = k then ... | Mathlib.CategoryTheory.Preadditive.Mat.332_0.xG9GKY7NTklnF73 | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom | Mathlib_CategoryTheory_Preadditive_Mat |
case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
h : ¬j = i
⊢ 0 = OfNat.ofNat 0 PUnit.unit PUnit.unit | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | rfl | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom := biproduct.lift fun i j k => if h : j = i then eqToHom (congr_arg M.X h) else 0
inv := biproduct.desc fun i j k => if h : i = k then ... | Mathlib.CategoryTheory.Preadditive.Mat.332_0.xG9GKY7NTklnF73 | /-- Every object in `Mat_ C` is isomorphic to the biproduct of its summands.
-/
@[simps]
def isoBiproductEmbedding (M : Mat_ C) : M ≅ ⨁ fun i => (embedding C).obj (M.X i) where
hom | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Preadditive C
D : Type u₁
inst✝² : Category.{v₁, u₁} D
inst✝¹ : Preadditive D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M : Mat_ C
i : M.ι
⊢ (additiveObjIsoBiproduct F M).hom ≫ biproduct.π (fun i => F.obj ((embedding C).obj (X M i))) i =
F.map ((isoBiproductEmbeddin... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | dsimp [additiveObjIsoBiproduct] | @[reassoc (attr := simp)]
lemma additiveObjIsoBiproduct_hom_π (F : Mat_ C ⥤ D) [Functor.Additive F] (M : Mat_ C) (i : M.ι) :
(additiveObjIsoBiproduct F M).hom ≫ biproduct.π _ i =
F.map (M.isoBiproductEmbedding.hom ≫ biproduct.π _ i) := by
| Mathlib.CategoryTheory.Preadditive.Mat.384_0.xG9GKY7NTklnF73 | @[reassoc (attr | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Preadditive C
D : Type u₁
inst✝² : Category.{v₁, u₁} D
inst✝¹ : Preadditive D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M : Mat_ C
i : M.ι
⊢ (F.map (biproduct.lift fun i j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) ≫
biproduct.lift (Functor.mapBi... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | rw [biproduct.lift_π, Category.assoc] | @[reassoc (attr := simp)]
lemma additiveObjIsoBiproduct_hom_π (F : Mat_ C ⥤ D) [Functor.Additive F] (M : Mat_ C) (i : M.ι) :
(additiveObjIsoBiproduct F M).hom ≫ biproduct.π _ i =
F.map (M.isoBiproductEmbedding.hom ≫ biproduct.π _ i) := by
dsimp [additiveObjIsoBiproduct]
| Mathlib.CategoryTheory.Preadditive.Mat.384_0.xG9GKY7NTklnF73 | @[reassoc (attr | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Preadditive C
D : Type u₁
inst✝² : Category.{v₁, u₁} D
inst✝¹ : Preadditive D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M : Mat_ C
i : M.ι
⊢ F.map (biproduct.lift fun i j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) ≫
biproduct.lift (Functor.mapBicon... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | erw [biproduct.lift_π, ← F.map_comp] | @[reassoc (attr := simp)]
lemma additiveObjIsoBiproduct_hom_π (F : Mat_ C ⥤ D) [Functor.Additive F] (M : Mat_ C) (i : M.ι) :
(additiveObjIsoBiproduct F M).hom ≫ biproduct.π _ i =
F.map (M.isoBiproductEmbedding.hom ≫ biproduct.π _ i) := by
dsimp [additiveObjIsoBiproduct]
rw [biproduct.lift_π, Category.asso... | Mathlib.CategoryTheory.Preadditive.Mat.384_0.xG9GKY7NTklnF73 | @[reassoc (attr | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Preadditive C
D : Type u₁
inst✝² : Category.{v₁, u₁} D
inst✝¹ : Preadditive D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M : Mat_ C
i : M.ι
⊢ F.map
((biproduct.lift fun i j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) ≫
Bicone.π (biproduct.bic... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp | @[reassoc (attr := simp)]
lemma additiveObjIsoBiproduct_hom_π (F : Mat_ C ⥤ D) [Functor.Additive F] (M : Mat_ C) (i : M.ι) :
(additiveObjIsoBiproduct F M).hom ≫ biproduct.π _ i =
F.map (M.isoBiproductEmbedding.hom ≫ biproduct.π _ i) := by
dsimp [additiveObjIsoBiproduct]
rw [biproduct.lift_π, Category.asso... | Mathlib.CategoryTheory.Preadditive.Mat.384_0.xG9GKY7NTklnF73 | @[reassoc (attr | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Preadditive C
D : Type u₁
inst✝² : Category.{v₁, u₁} D
inst✝¹ : Preadditive D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M : Mat_ C
i : M.ι
⊢ biproduct.ι (fun i => F.obj ((embedding C).obj (X M i))) i ≫ (additiveObjIsoBiproduct F M).inv =
F.map (biproduct.ι (fun i =>... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | dsimp [additiveObjIsoBiproduct, Functor.mapBiproduct, Functor.mapBicone] | @[reassoc (attr := simp)]
lemma ι_additiveObjIsoBiproduct_inv (F : Mat_ C ⥤ D) [Functor.Additive F] (M : Mat_ C) (i : M.ι) :
biproduct.ι _ i ≫ (additiveObjIsoBiproduct F M).inv =
F.map (biproduct.ι _ i ≫ M.isoBiproductEmbedding.inv) := by
| Mathlib.CategoryTheory.Preadditive.Mat.393_0.xG9GKY7NTklnF73 | @[reassoc (attr | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Preadditive C
D : Type u₁
inst✝² : Category.{v₁, u₁} D
inst✝¹ : Preadditive D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M : Mat_ C
i : M.ι
⊢ biproduct.ι (fun i => F.obj ((embedding C).obj (X M i))) i ≫
(biproduct.desc fun j => F.map (biproduct.ι (fun i => (embeddi... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp only [biproduct.ι_desc, biproduct.ι_desc_assoc, ← F.map_comp] | @[reassoc (attr := simp)]
lemma ι_additiveObjIsoBiproduct_inv (F : Mat_ C ⥤ D) [Functor.Additive F] (M : Mat_ C) (i : M.ι) :
biproduct.ι _ i ≫ (additiveObjIsoBiproduct F M).inv =
F.map (biproduct.ι _ i ≫ M.isoBiproductEmbedding.inv) := by
dsimp [additiveObjIsoBiproduct, Functor.mapBiproduct, Functor.mapBico... | Mathlib.CategoryTheory.Preadditive.Mat.393_0.xG9GKY7NTklnF73 | @[reassoc (attr | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M N : Mat_ C
f : M ⟶ N
⊢ F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫ biproduc... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | ext i : 1 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) := by
| Mathlib.CategoryTheory.Preadditive.Mat.402_0.xG9GKY7NTklnF73 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) | Mathlib_CategoryTheory_Preadditive_Mat |
case w
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M N : Mat_ C
f : M ⟶ N
i : N.ι
⊢ (F.map f ≫ (additiveObjIsoBiproduct F N).hom) ≫ biproduct.π (fun i => F.obj ((e... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp only [Category.assoc, additiveObjIsoBiproduct_hom_π, isoBiproductEmbedding_hom,
embedding_obj_ι, embedding_obj_X, biproduct.lift_π, biproduct.matrix_π,
← cancel_epi (additiveObjIsoBiproduct F M).inv, Iso.inv_hom_id_assoc] | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) := by
ext i : 1
| Mathlib.CategoryTheory.Preadditive.Mat.402_0.xG9GKY7NTklnF73 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) | Mathlib_CategoryTheory_Preadditive_Mat |
case w
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M N : Mat_ C
f : M ⟶ N
i : N.ι
⊢ ((additiveObjIsoBiproduct F M).inv ≫
F.map f ≫ F.map fun j k => if h : j ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | ext j : 1 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) := by
ext i : 1
simp only [Ca... | Mathlib.CategoryTheory.Preadditive.Mat.402_0.xG9GKY7NTklnF73 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) | Mathlib_CategoryTheory_Preadditive_Mat |
case w.w
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M N : Mat_ C
f : M ⟶ N
i : N.ι
j : M.ι
⊢ (biproduct.ι (fun i => F.obj ((embedding C).obj (X M i))) j ≫
(... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp only [ι_additiveObjIsoBiproduct_inv_assoc, isoBiproductEmbedding_inv,
biproduct.ι_desc, ← F.map_comp] | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) := by
ext i : 1
simp only [Ca... | Mathlib.CategoryTheory.Preadditive.Mat.402_0.xG9GKY7NTklnF73 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) | Mathlib_CategoryTheory_Preadditive_Mat |
case w.w
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M N : Mat_ C
f : M ⟶ N
i : N.ι
j : M.ι
⊢ F.map
((fun j_1 k => if h : j = k then eqToHom (_ : X M j = X M... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | congr 1 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) := by
ext i : 1
simp only [Ca... | Mathlib.CategoryTheory.Preadditive.Mat.402_0.xG9GKY7NTklnF73 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) | Mathlib_CategoryTheory_Preadditive_Mat |
case w.w.e_a
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M N : Mat_ C
f : M ⟶ N
i : N.ι
j : M.ι
⊢ ((fun j_1 k => if h : j = k then eqToHom (_ : X M j = X M k) else... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | funext ⟨⟩ ⟨⟩ | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) := by
ext i : 1
simp only [Ca... | Mathlib.CategoryTheory.Preadditive.Mat.402_0.xG9GKY7NTklnF73 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) | Mathlib_CategoryTheory_Preadditive_Mat |
case w.w.e_a.h.h
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M N : Mat_ C
f : M ⟶ N
i : N.ι
j : M.ι
⊢ ((fun j_1 k => if h : j = k then eqToHom (_ : X M j = X M k) ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp [comp_apply, dite_comp, comp_dite] | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) := by
ext i : 1
simp only [Ca... | Mathlib.CategoryTheory.Preadditive.Mat.402_0.xG9GKY7NTklnF73 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
F.map f ≫ (additiveObjIsoBiproduct F N).hom =
(additiveObjIsoBiproduct F M).hom ≫
biproduct.matrix fun i j => F.map ((embedding C).map (f i j)) | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : Functor.Additive F
M N : Mat_ C
f : M ⟶ N
⊢ (additiveObjIsoBiproduct F M).inv ≫ F.map f =
(biproduct.matrix fun i j => F.map ((embeddi... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | rw [Iso.inv_comp_eq, ← Category.assoc, Iso.eq_comp_inv, additiveObjIsoBiproduct_naturality] | @[reassoc]
theorem additiveObjIsoBiproduct_naturality' (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
(additiveObjIsoBiproduct F M).inv ≫ F.map f =
biproduct.matrix (fun i j => F.map ((embedding C).map (f i j)) : _) ≫
(additiveObjIsoBiproduct F N).inv :=
by | Mathlib.CategoryTheory.Preadditive.Mat.421_0.xG9GKY7NTklnF73 | @[reassoc]
theorem additiveObjIsoBiproduct_naturality' (F : Mat_ C ⥤ D) [Functor.Additive F] {M N : Mat_ C}
(f : M ⟶ N) :
(additiveObjIsoBiproduct F M).inv ≫ F.map f =
biproduct.matrix (fun i j => F.map ((embedding C).map (f i j)) : _) ≫
(additiveObjIsoBiproduct F N).inv | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : C ⥤ D
inst✝ : Functor.Additive F
X : Mat_ C
⊢ { obj := fun X => ⨁ fun i => F.obj (CategoryTheory.Mat_.X X i),
map := fun {X Y} f => biproduct.matri... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | dsimp | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X := ⨁ fun i => F.obj (X.X i)
map f := biproduct.matrix fun i j => F.map (f i j)
map_id X := by
| Mathlib.CategoryTheory.Preadditive.Mat.433_0.xG9GKY7NTklnF73 | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : C ⥤ D
inst✝ : Functor.Additive F
X : Mat_ C
⊢ (biproduct.matrix fun i j => F.map (𝟙 X i j)) = 𝟙 (⨁ fun i => F.obj (CategoryTheory.Mat_.X X i)) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | ext i j | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X := ⨁ fun i => F.obj (X.X i)
map f := biproduct.matrix fun i j => F.map (f i j)
map_id X := by
dsimp
| Mathlib.CategoryTheory.Preadditive.Mat.433_0.xG9GKY7NTklnF73 | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case w.w
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : C ⥤ D
inst✝ : Functor.Additive F
X : Mat_ C
i j : X.ι
⊢ biproduct.ι (fun i => F.obj (CategoryTheory.Mat_.X X i)) j ≫
(biproduct.matrix fun i j... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | by_cases h : j = i | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X := ⨁ fun i => F.obj (X.X i)
map f := biproduct.matrix fun i j => F.map (f i j)
map_id X := by
dsimp
ext i j
| Mathlib.CategoryTheory.Preadditive.Mat.433_0.xG9GKY7NTklnF73 | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : C ⥤ D
inst✝ : Functor.Additive F
X : Mat_ C
i j : X.ι
h : j = i
⊢ biproduct.ι (fun i => F.obj (CategoryTheory.Mat_.X X i)) j ≫
(biproduct.matr... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | subst h | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X := ⨁ fun i => F.obj (X.X i)
map f := biproduct.matrix fun i j => F.map (f i j)
map_id X := by
dsimp
ext i j
by_cas... | Mathlib.CategoryTheory.Preadditive.Mat.433_0.xG9GKY7NTklnF73 | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : C ⥤ D
inst✝ : Functor.Additive F
X : Mat_ C
j : X.ι
⊢ biproduct.ι (fun i => F.obj (CategoryTheory.Mat_.X X i)) j ≫
(biproduct.matrix fun i j =... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X := ⨁ fun i => F.obj (X.X i)
map f := biproduct.matrix fun i j => F.map (f i j)
map_id X := by
dsimp
ext i j
by_cas... | Mathlib.CategoryTheory.Preadditive.Mat.433_0.xG9GKY7NTklnF73 | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case neg
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : C ⥤ D
inst✝ : Functor.Additive F
X : Mat_ C
i j : X.ι
h : ¬j = i
⊢ biproduct.ι (fun i => F.obj (CategoryTheory.Mat_.X X i)) j ≫
(biproduct.mat... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp [h] | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X := ⨁ fun i => F.obj (X.X i)
map f := biproduct.matrix fun i j => F.map (f i j)
map_id X := by
dsimp
ext i j
by_cas... | Mathlib.CategoryTheory.Preadditive.Mat.433_0.xG9GKY7NTklnF73 | /-- Any additive functor `C ⥤ D` to a category `D` with finite biproducts extends to
a functor `Mat_ C ⥤ D`. -/
@[simps]
def lift (F : C ⥤ D) [Functor.Additive F] : Mat_ C ⥤ D where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
⊢ L.map f ≫
((fun M =>
... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | dsimp only [Iso.trans_hom, Iso.symm_hom, biproduct.mapIso_hom] | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
⊢ L.map f ≫
(additiveObj... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp only [additiveObjIsoBiproduct_naturality_assoc] | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
⊢ (additiveObjIsoBiproduct L X... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp only [biproduct.matrix_map_assoc, Category.assoc] | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
⊢ (additiveObjIsoBiproduct L X... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp only [additiveObjIsoBiproduct_naturality'] | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
⊢ (additiveObjIsoBiproduct L X... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp only [biproduct.map_matrix_assoc, Category.assoc] | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
⊢ (additiveObjIsoBiproduct L X... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | congr 3 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
case e_a.e_a.e_m
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
⊢ (fun j k =>... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | ext j k | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
case e_a.e_a.e_m.h.h
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
j : X✝.ι
... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | apply biproduct.hom_ext | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
case e_a.e_a.e_m.h.h.w
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
j : X✝.... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | rintro ⟨⟩ | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
case e_a.e_a.e_m.h.h.w.unit
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
j ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | dsimp | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
case e_a.e_a.e_m.h.h.w.unit
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₁
inst✝⁴ : Category.{v₁, u₁} D
inst✝³ : Preadditive D
inst✝² : HasFiniteBiproducts D
F : C ⥤ D
inst✝¹ : Functor.Additive F
L : Mat_ C ⥤ D
inst✝ : Functor.Additive L
α : embedding C ⋙ L ≅ F
X✝ Y✝ : Mat_ C
f : X✝ ⟶ Y✝
j ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simpa using α.hom.naturality (f j k) | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
(fun M =>
additiveObjIsoBiproduct L M ≪≫
(bipr... | Mathlib.CategoryTheory.Preadditive.Mat.462_0.xG9GKY7NTklnF73 | /-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
R : Type u
⊢ Inhabited (Mat R) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | dsimp [Mat] | instance (R : Type u) : Inhabited (Mat R) := by
| Mathlib.CategoryTheory.Preadditive.Mat.551_0.xG9GKY7NTklnF73 | instance (R : Type u) : Inhabited (Mat R) | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
R : Type u
⊢ Inhabited FintypeCat | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | infer_instance | instance (R : Type u) : Inhabited (Mat R) := by
dsimp [Mat]
| Mathlib.CategoryTheory.Preadditive.Mat.551_0.xG9GKY7NTklnF73 | instance (R : Type u) : Inhabited (Mat R) | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type u
inst✝ : Semiring R
⊢ ∀ {W X Y Z : Mat R} (f : W ⟶ X) (g : X ⟶ Y) (h : Y ⟶ Z), (f ≫ g) ≫ h = f ≫ g ≫ h | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | intros | instance (R : Type u) [Semiring R] : Category (Mat R) where
Hom X Y := Matrix X Y R
id X := (1 : Matrix X X R)
comp {X Y Z} f g := (show Matrix X Y R from f) * (show Matrix Y Z R from g)
assoc := by | Mathlib.CategoryTheory.Preadditive.Mat.560_0.xG9GKY7NTklnF73 | instance (R : Type u) [Semiring R] : Category (Mat R) where
Hom X Y | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type u
inst✝ : Semiring R
W✝ X✝ Y✝ Z✝ : Mat R
f✝ : W✝ ⟶ X✝
g✝ : X✝ ⟶ Y✝
h✝ : Y✝ ⟶ Z✝
⊢ (f✝ ≫ g✝) ≫ h✝ = f✝ ≫ g✝ ≫ h✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp [Matrix.mul_assoc] | instance (R : Type u) [Semiring R] : Category (Mat R) where
Hom X Y := Matrix X Y R
id X := (1 : Matrix X X R)
comp {X Y Z} f g := (show Matrix X Y R from f) * (show Matrix Y Z R from g)
assoc := by intros; | Mathlib.CategoryTheory.Preadditive.Mat.560_0.xG9GKY7NTklnF73 | instance (R : Type u) [Semiring R] : Category (Mat R) where
Hom X Y | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type u
inst✝ : Semiring R
M : Mat R
i : ↑M
⊢ 𝟙 M i i = 1 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp [id_apply] | @[simp]
theorem id_apply_self (M : Mat R) (i : M) : (𝟙 M : Matrix M M R) i i = 1 := by | Mathlib.CategoryTheory.Preadditive.Mat.590_0.xG9GKY7NTklnF73 | @[simp]
theorem id_apply_self (M : Mat R) (i : M) : (𝟙 M : Matrix M M R) i i = 1 | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type u
inst✝ : Semiring R
M : Mat R
i j : ↑M
h : i ≠ j
⊢ 𝟙 M i j = 0 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp [id_apply, h] | @[simp]
theorem id_apply_of_ne (M : Mat R) (i j : M) (h : i ≠ j) : (𝟙 M : Matrix M M R) i j = 0 := by
| Mathlib.CategoryTheory.Preadditive.Mat.595_0.xG9GKY7NTklnF73 | @[simp]
theorem id_apply_of_ne (M : Mat R) (i j : M) (h : i ≠ j) : (𝟙 M : Matrix M M R) i j = 0 | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X : Mat_ (SingleObj Rᵐᵒᵖ)
⊢ { obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => MulOpposite.unop (f i j) }.map (𝟙 X) =
𝟙 ({ obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => MulOpposite.unop (f i j) }.obj... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | ext | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X := FintypeCat.of X.ι
map f i j := MulOpposite.unop (f i j)
map_id X := by
| Mathlib.CategoryTheory.Preadditive.Mat.623_0.xG9GKY7NTklnF73 | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X : Mat_ (SingleObj Rᵐᵒᵖ)
i✝ j✝ : ↑({ obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => MulOpposite.unop (f i j) }.obj X)
⊢ { obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => MulOpposite.unop (f i j) }.... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp only [Mat_.id_def, id_def] | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X := FintypeCat.of X.ι
map f i j := MulOpposite.unop (f i j)
map_id X := by
ext
| Mathlib.CategoryTheory.Preadditive.Mat.623_0.xG9GKY7NTklnF73 | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X : Mat_ (SingleObj Rᵐᵒᵖ)
i✝ j✝ : ↑({ obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => MulOpposite.unop (f i j) }.obj X)
⊢ MulOpposite.unop (if h : i✝ = j✝ then eqToHom (_ : Mat_.X X i✝ = Mat_.X X j✝) else 0) = if... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | split_ifs | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X := FintypeCat.of X.ι
map f i j := MulOpposite.unop (f i j)
map_id X := by
ext
simp only [Mat_.id_def, id_def]
| Mathlib.CategoryTheory.Preadditive.Mat.623_0.xG9GKY7NTklnF73 | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X : Mat_ (SingleObj Rᵐᵒᵖ)
i✝ j✝ : ↑({ obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => MulOpposite.unop (f i j) }.obj X)
h✝ : i✝ = j✝
⊢ MulOpposite.unop (eqToHom (_ : Mat_.X X i✝ = Mat_.X X j✝)) = 1 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | rfl | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X := FintypeCat.of X.ι
map f i j := MulOpposite.unop (f i j)
map_id X := by
ext
simp only [Mat_.id_def, id_def]
split_ifs <;> | Mathlib.CategoryTheory.Preadditive.Mat.623_0.xG9GKY7NTklnF73 | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case neg
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X : Mat_ (SingleObj Rᵐᵒᵖ)
i✝ j✝ : ↑({ obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => MulOpposite.unop (f i j) }.obj X)
h✝ : ¬i✝ = j✝
⊢ MulOpposite.unop 0 = 0 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | rfl | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X := FintypeCat.of X.ι
map f i j := MulOpposite.unop (f i j)
map_id X := by
ext
simp only [Mat_.id_def, id_def]
split_ifs <;> | Mathlib.CategoryTheory.Preadditive.Mat.623_0.xG9GKY7NTklnF73 | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X✝ Y✝ Z✝ : Mat_ (SingleObj Rᵐᵒᵖ)
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ { obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => MulOpposite.unop (f i j) }.map (f ≫ g) =
{ obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => Mu... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | ext | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X := FintypeCat.of X.ι
map f i j := MulOpposite.unop (f i j)
map_id X := by
ext
simp only [Mat_.id_def, id_def]
split_ifs <;> rfl
map_comp f g... | Mathlib.CategoryTheory.Preadditive.Mat.623_0.xG9GKY7NTklnF73 | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X✝ Y✝ Z✝ : Mat_ (SingleObj Rᵐᵒᵖ)
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
i✝ : ↑({ obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => MulOpposite.unop (f i j) }.obj X✝)
j✝ : ↑({ obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | simp only [Mat_.comp_apply, comp_apply] | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X := FintypeCat.of X.ι
map f i j := MulOpposite.unop (f i j)
map_id X := by
ext
simp only [Mat_.id_def, id_def]
split_ifs <;> rfl
map_comp f g... | Mathlib.CategoryTheory.Preadditive.Mat.623_0.xG9GKY7NTklnF73 | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X✝ Y✝ Z✝ : Mat_ (SingleObj Rᵐᵒᵖ)
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
i✝ : ↑({ obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f i j => MulOpposite.unop (f i j) }.obj X✝)
j✝ : ↑({ obj := fun X => FintypeCat.of X.ι, map := fun {X Y} f ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | apply Finset.unop_sum | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X := FintypeCat.of X.ι
map f i j := MulOpposite.unop (f i j)
map_id X := by
ext
simp only [Mat_.id_def, id_def]
split_ifs <;> rfl
map_comp f g... | Mathlib.CategoryTheory.Preadditive.Mat.623_0.xG9GKY7NTklnF73 | /-- Auxiliary definition for `CategoryTheory.Mat.equivalenceSingleObj`. -/
@[simps]
def equivalenceSingleObjInverse : Mat_ (SingleObj Rᵐᵒᵖ) ⥤ Mat R where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X✝ Y✝ : Mat_ (SingleObj Rᵐᵒᵖ)
a₁✝ a₂✝ : X✝ ⟶ Y✝
w : (equivalenceSingleObjInverse R).map a₁✝ = (equivalenceSingleObjInverse R).map a₂✝
⊢ a₁✝ = a₂✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | ext | instance : Faithful (equivalenceSingleObjInverse R) where
map_injective w := by
| Mathlib.CategoryTheory.Preadditive.Mat.640_0.xG9GKY7NTklnF73 | instance : Faithful (equivalenceSingleObjInverse R) where
map_injective w | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X✝ Y✝ : Mat_ (SingleObj Rᵐᵒᵖ)
a₁✝ a₂✝ : X✝ ⟶ Y✝
w : (equivalenceSingleObjInverse R).map a₁✝ = (equivalenceSingleObjInverse R).map a₂✝
i✝ : X✝.ι
j✝ : Y✝.ι
⊢ a₁✝ i✝ j✝ = a₂✝ i✝ j✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | apply_fun MulOpposite.unop using MulOpposite.unop_injective | instance : Faithful (equivalenceSingleObjInverse R) where
map_injective w := by
ext
| Mathlib.CategoryTheory.Preadditive.Mat.640_0.xG9GKY7NTklnF73 | instance : Faithful (equivalenceSingleObjInverse R) where
map_injective w | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X✝ Y✝ : Mat_ (SingleObj Rᵐᵒᵖ)
a₁✝ a₂✝ : X✝ ⟶ Y✝
w : (equivalenceSingleObjInverse R).map a₁✝ = (equivalenceSingleObjInverse R).map a₂✝
i✝ : X✝.ι
j✝ : Y✝.ι
⊢ MulOpposite.unop (a₁✝ i✝ j✝) = MulOpposite.unop (a₂✝ i✝ j✝) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | exact congr_fun (congr_fun w _) _ | instance : Faithful (equivalenceSingleObjInverse R) where
map_injective w := by
ext
apply_fun MulOpposite.unop using MulOpposite.unop_injective
| Mathlib.CategoryTheory.Preadditive.Mat.640_0.xG9GKY7NTklnF73 | instance : Faithful (equivalenceSingleObjInverse R) where
map_injective w | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X : Mat R
⊢ (equivalenceSingleObjInverse R).obj (Mat_.mk ↑X fun x => PUnit.unit) = X | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | dsimp | instance : EssSurj (equivalenceSingleObjInverse R) where
mem_essImage X :=
⟨{ ι := X
X := fun _ => PUnit.unit }, ⟨eqToIso (by | Mathlib.CategoryTheory.Preadditive.Mat.649_0.xG9GKY7NTklnF73 | instance : EssSurj (equivalenceSingleObjInverse R) where
mem_essImage X | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X : Mat R
⊢ FintypeCat.of ↑X = X | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | cases X | instance : EssSurj (equivalenceSingleObjInverse R) where
mem_essImage X :=
⟨{ ι := X
X := fun _ => PUnit.unit }, ⟨eqToIso (by dsimp; | Mathlib.CategoryTheory.Preadditive.Mat.649_0.xG9GKY7NTklnF73 | instance : EssSurj (equivalenceSingleObjInverse R) where
mem_essImage X | Mathlib_CategoryTheory_Preadditive_Mat |
case mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
α✝ : Type
str✝ : Fintype α✝
⊢ FintypeCat.of ↑(Bundled.mk α✝) = Bundled.mk α✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | congr | instance : EssSurj (equivalenceSingleObjInverse R) where
mem_essImage X :=
⟨{ ι := X
X := fun _ => PUnit.unit }, ⟨eqToIso (by dsimp; cases X; | Mathlib.CategoryTheory.Preadditive.Mat.649_0.xG9GKY7NTklnF73 | instance : EssSurj (equivalenceSingleObjInverse R) where
mem_essImage X | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X Y : Mat R
⊢ AddCommGroup (X ⟶ Y) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | change AddCommGroup (Matrix X Y R) | instance (X Y : Mat R) : AddCommGroup (X ⟶ Y) := by
| Mathlib.CategoryTheory.Preadditive.Mat.663_0.xG9GKY7NTklnF73 | instance (X Y : Mat R) : AddCommGroup (X ⟶ Y) | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
R : Type
inst✝ : Ring R
X Y : Mat R
⊢ AddCommGroup (Matrix (↑X) (↑Y) R) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.... | infer_instance | instance (X Y : Mat R) : AddCommGroup (X ⟶ Y) := by
change AddCommGroup (Matrix X Y R)
| Mathlib.CategoryTheory.Preadditive.Mat.663_0.xG9GKY7NTklnF73 | instance (X Y : Mat R) : AddCommGroup (X ⟶ Y) | Mathlib_CategoryTheory_Preadditive_Mat |
ι : Type u_1
I J : Box ι
x y : ι → ℝ
⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | tfae_have 1 ↔ 2 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
| Mathlib.Analysis.BoxIntegral.Box.Basic.151_0.huZHUvFo9Qi2rb8 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | Mathlib_Analysis_BoxIntegral_Box_Basic |
case tfae_1_iff_2
ι : Type u_1
I J : Box ι
x y : ι → ℝ
⊢ I ≤ J ↔ ↑I ⊆ ↑J
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | exact Iff.rfl | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
tfae_have 1 ↔ 2; | Mathlib.Analysis.BoxIntegral.Box.Basic.151_0.huZHUvFo9Qi2rb8 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | tfae_have 2 → 3 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
tfae_have 1 ↔ 2; exact Iff.rfl
| Mathlib.Analysis.BoxIntegral.Box.Basic.151_0.huZHUvFo9Qi2rb8 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | Mathlib_Analysis_BoxIntegral_Box_Basic |
case tfae_2_to_3
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
⊢ ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | intro h | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
tfae_have 1 ↔ 2; exact Iff.rfl
tfae_have 2 → 3
· | Mathlib.Analysis.BoxIntegral.Box.Basic.151_0.huZHUvFo9Qi2rb8 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | Mathlib_Analysis_BoxIntegral_Box_Basic |
case tfae_2_to_3
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
h : ↑I ⊆ ↑J
⊢ Icc I.lower I.upper ⊆ Icc J.lower J.upper | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
tfae_have 1 ↔ 2; exact Iff.rfl
tfae_have 2 → 3
· intro h
| Mathlib.Analysis.BoxIntegral.Box.Basic.151_0.huZHUvFo9Qi2rb8 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | tfae_have 3 ↔ 4 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
tfae_have 1 ↔ 2; exact Iff.rfl
tfae_have 2 → 3
· intro h
simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h
| Mathlib.Analysis.BoxIntegral.Box.Basic.151_0.huZHUvFo9Qi2rb8 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | Mathlib_Analysis_BoxIntegral_Box_Basic |
case tfae_3_iff_4
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
⊢ Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | exact Icc_subset_Icc_iff I.lower_le_upper | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
tfae_have 1 ↔ 2; exact Iff.rfl
tfae_have 2 → 3
· intro h
simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h
tfae_have 3 ↔ 4; | Mathlib.Analysis.BoxIntegral.Box.Basic.151_0.huZHUvFo9Qi2rb8 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
tfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | tfae_have 4 → 2 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
tfae_have 1 ↔ 2; exact Iff.rfl
tfae_have 2 → 3
· intro h
simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h
tfae_have 3 ↔ 4; exact Icc_su... | Mathlib.Analysis.BoxIntegral.Box.Basic.151_0.huZHUvFo9Qi2rb8 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | Mathlib_Analysis_BoxIntegral_Box_Basic |
case tfae_4_to_2
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
tfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
⊢ J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J
ι : Type u_1
I J : Box... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | exact fun h x hx i ↦ Ioc_subset_Ioc (h.1 i) (h.2 i) (hx i) | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
tfae_have 1 ↔ 2; exact Iff.rfl
tfae_have 2 → 3
· intro h
simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h
tfae_have 3 ↔ 4; exact Icc_su... | Mathlib.Analysis.BoxIntegral.Box.Basic.151_0.huZHUvFo9Qi2rb8 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y : ι → ℝ
tfae_1_iff_2 : I ≤ J ↔ ↑I ⊆ ↑J
tfae_2_to_3 : ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper
tfae_3_iff_4 : Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.upper
tfae_4_to_2 : J.lower ≤ I.lower ∧ I.upper ≤ J.upper → ↑I ⊆ ↑J
⊢ List.TFAE [I ≤ J, ↑I ⊆ ↑J... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | tfae_finish | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
tfae_have 1 ↔ 2; exact Iff.rfl
tfae_have 2 → 3
· intro h
simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h
tfae_have 3 ↔ 4; exact Icc_su... | Mathlib.Analysis.BoxIntegral.Box.Basic.151_0.huZHUvFo9Qi2rb8 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y : ι → ℝ
⊢ Injective toSet | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | rintro ⟨l₁, u₁, h₁⟩ ⟨l₂, u₂, h₂⟩ h | theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) := by
| Mathlib.Analysis.BoxIntegral.Box.Basic.172_0.huZHUvFo9Qi2rb8 | theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) | Mathlib_Analysis_BoxIntegral_Box_Basic |
case mk.mk
ι : Type u_1
I J : Box ι
x y l₁ u₁ : ι → ℝ
h₁ : ∀ (i : ι), l₁ i < u₁ i
l₂ u₂ : ι → ℝ
h₂ : ∀ (i : ι), l₂ i < u₂ i
h : ↑{ lower := l₁, upper := u₁, lower_lt_upper := h₁ } = ↑{ lower := l₂, upper := u₂, lower_lt_upper := h₂ }
⊢ { lower := l₁, upper := u₁, lower_lt_upper := h₁ } = { lower := l₂, upper := u₂, low... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simp only [Subset.antisymm_iff, coe_subset_coe, le_iff_bounds] at h | theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) := by
rintro ⟨l₁, u₁, h₁⟩ ⟨l₂, u₂, h₂⟩ h
| Mathlib.Analysis.BoxIntegral.Box.Basic.172_0.huZHUvFo9Qi2rb8 | theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) | Mathlib_Analysis_BoxIntegral_Box_Basic |
case mk.mk
ι : Type u_1
I J : Box ι
x y l₁ u₁ : ι → ℝ
h₁ : ∀ (i : ι), l₁ i < u₁ i
l₂ u₂ : ι → ℝ
h₂ : ∀ (i : ι), l₂ i < u₂ i
h : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁
⊢ { lower := l₁, upper := u₁, lower_lt_upper := h₁ } = { lower := l₂, upper := u₂, lower_lt_upper := h₂ } | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | congr | theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) := by
rintro ⟨l₁, u₁, h₁⟩ ⟨l₂, u₂, h₂⟩ h
simp only [Subset.antisymm_iff, coe_subset_coe, le_iff_bounds] at h
| Mathlib.Analysis.BoxIntegral.Box.Basic.172_0.huZHUvFo9Qi2rb8 | theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) | Mathlib_Analysis_BoxIntegral_Box_Basic |
case mk.mk.e_lower
ι : Type u_1
I J : Box ι
x y l₁ u₁ : ι → ℝ
h₁ : ∀ (i : ι), l₁ i < u₁ i
l₂ u₂ : ι → ℝ
h₂ : ∀ (i : ι), l₂ i < u₂ i
h : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁
⊢ l₁ = l₂
case mk.mk.e_upper
ι : Type u_1
I J : Box ι
x y l₁ u₁ : ι → ℝ
h₁ : ∀ (i : ι), l₁ i < u₁ i
l₂ u₂ : ι → ℝ
h₂ : ∀ (i : ι), l₂ i < u₂ i
h ... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | exacts [le_antisymm h.2.1 h.1.1, le_antisymm h.1.2 h.2.2] | theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) := by
rintro ⟨l₁, u₁, h₁⟩ ⟨l₂, u₂, h₂⟩ h
simp only [Subset.antisymm_iff, coe_subset_coe, le_iff_bounds] at h
congr
| Mathlib.Analysis.BoxIntegral.Box.Basic.172_0.huZHUvFo9Qi2rb8 | theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y : ι → ℝ
⊢ Option.isSome ⊥ = true ↔ Set.Nonempty ↑⊥ | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | erw [Option.isSome] | theorem isSome_iff : ∀ {I : WithBot (Box ι)}, I.isSome ↔ (I : Set (ι → ℝ)).Nonempty
| ⊥ => by
| Mathlib.Analysis.BoxIntegral.Box.Basic.280_0.huZHUvFo9Qi2rb8 | theorem isSome_iff : ∀ {I : WithBot (Box ι)}, I.isSome ↔ (I : Set (ι → ℝ)).Nonempty
| ⊥ => by
erw [Option.isSome]
simp
| (I : Box ι) => by
erw [Option.isSome]
simp [I.nonempty_coe] | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y : ι → ℝ
⊢ (match ⊥ with
| some val => true
| none => false) =
true ↔
Set.Nonempty ↑⊥ | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simp | theorem isSome_iff : ∀ {I : WithBot (Box ι)}, I.isSome ↔ (I : Set (ι → ℝ)).Nonempty
| ⊥ => by
erw [Option.isSome]
| Mathlib.Analysis.BoxIntegral.Box.Basic.280_0.huZHUvFo9Qi2rb8 | theorem isSome_iff : ∀ {I : WithBot (Box ι)}, I.isSome ↔ (I : Set (ι → ℝ)).Nonempty
| ⊥ => by
erw [Option.isSome]
simp
| (I : Box ι) => by
erw [Option.isSome]
simp [I.nonempty_coe] | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I✝ J : Box ι
x y : ι → ℝ
I : Box ι
⊢ Option.isSome ↑I = true ↔ Set.Nonempty ↑↑I | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | erw [Option.isSome] | theorem isSome_iff : ∀ {I : WithBot (Box ι)}, I.isSome ↔ (I : Set (ι → ℝ)).Nonempty
| ⊥ => by
erw [Option.isSome]
simp
| (I : Box ι) => by
| Mathlib.Analysis.BoxIntegral.Box.Basic.280_0.huZHUvFo9Qi2rb8 | theorem isSome_iff : ∀ {I : WithBot (Box ι)}, I.isSome ↔ (I : Set (ι → ℝ)).Nonempty
| ⊥ => by
erw [Option.isSome]
simp
| (I : Box ι) => by
erw [Option.isSome]
simp [I.nonempty_coe] | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I✝ J : Box ι
x y : ι → ℝ
I : Box ι
⊢ (match ↑I with
| some val => true
| none => false) =
true ↔
Set.Nonempty ↑↑I | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simp [I.nonempty_coe] | theorem isSome_iff : ∀ {I : WithBot (Box ι)}, I.isSome ↔ (I : Set (ι → ℝ)).Nonempty
| ⊥ => by
erw [Option.isSome]
simp
| (I : Box ι) => by
erw [Option.isSome]
| Mathlib.Analysis.BoxIntegral.Box.Basic.280_0.huZHUvFo9Qi2rb8 | theorem isSome_iff : ∀ {I : WithBot (Box ι)}, I.isSome ↔ (I : Set (ι → ℝ)).Nonempty
| ⊥ => by
erw [Option.isSome]
simp
| (I : Box ι) => by
erw [Option.isSome]
simp [I.nonempty_coe] | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I✝ J : Box ι
x y : ι → ℝ
I : WithBot (Box ι)
⊢ ⋃ J, ⋃ (_ : ↑J = I), ↑J = ↑I | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | induction I using WithBot.recBotCoe | theorem biUnion_coe_eq_coe (I : WithBot (Box ι)) :
⋃ (J : Box ι) (_ : ↑J = I), (J : Set (ι → ℝ)) = I := by
| Mathlib.Analysis.BoxIntegral.Box.Basic.289_0.huZHUvFo9Qi2rb8 | theorem biUnion_coe_eq_coe (I : WithBot (Box ι)) :
⋃ (J : Box ι) (_ : ↑J = I), (J : Set (ι → ℝ)) = I | Mathlib_Analysis_BoxIntegral_Box_Basic |
case bot
ι : Type u_1
I J : Box ι
x y : ι → ℝ
⊢ ⋃ J, ⋃ (_ : ↑J = ⊥), ↑J = ↑⊥ | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simp [WithBot.coe_eq_coe] | theorem biUnion_coe_eq_coe (I : WithBot (Box ι)) :
⋃ (J : Box ι) (_ : ↑J = I), (J : Set (ι → ℝ)) = I := by
induction I using WithBot.recBotCoe <;> | Mathlib.Analysis.BoxIntegral.Box.Basic.289_0.huZHUvFo9Qi2rb8 | theorem biUnion_coe_eq_coe (I : WithBot (Box ι)) :
⋃ (J : Box ι) (_ : ↑J = I), (J : Set (ι → ℝ)) = I | Mathlib_Analysis_BoxIntegral_Box_Basic |
case coe
ι : Type u_1
I J : Box ι
x y : ι → ℝ
a✝ : Box ι
⊢ ⋃ J, ⋃ (_ : ↑J = ↑a✝), ↑J = ↑↑a✝ | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simp [WithBot.coe_eq_coe] | theorem biUnion_coe_eq_coe (I : WithBot (Box ι)) :
⋃ (J : Box ι) (_ : ↑J = I), (J : Set (ι → ℝ)) = I := by
induction I using WithBot.recBotCoe <;> | Mathlib.Analysis.BoxIntegral.Box.Basic.289_0.huZHUvFo9Qi2rb8 | theorem biUnion_coe_eq_coe (I : WithBot (Box ι)) :
⋃ (J : Box ι) (_ : ↑J = I), (J : Set (ι → ℝ)) = I | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I✝ J✝ : Box ι
x y : ι → ℝ
I J : WithBot (Box ι)
⊢ ↑I ⊆ ↑J ↔ I ≤ J | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | induction I using WithBot.recBotCoe | @[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J := by
| Mathlib.Analysis.BoxIntegral.Box.Basic.294_0.huZHUvFo9Qi2rb8 | @[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J | Mathlib_Analysis_BoxIntegral_Box_Basic |
case bot
ι : Type u_1
I J✝ : Box ι
x y : ι → ℝ
J : WithBot (Box ι)
⊢ ↑⊥ ⊆ ↑J ↔ ⊥ ≤ J | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simp | @[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J := by
induction I using WithBot.recBotCoe; · | Mathlib.Analysis.BoxIntegral.Box.Basic.294_0.huZHUvFo9Qi2rb8 | @[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J | Mathlib_Analysis_BoxIntegral_Box_Basic |
case coe
ι : Type u_1
I J✝ : Box ι
x y : ι → ℝ
J : WithBot (Box ι)
a✝ : Box ι
⊢ ↑↑a✝ ⊆ ↑J ↔ ↑a✝ ≤ J | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | induction J using WithBot.recBotCoe | @[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J := by
induction I using WithBot.recBotCoe; · simp
| Mathlib.Analysis.BoxIntegral.Box.Basic.294_0.huZHUvFo9Qi2rb8 | @[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J | Mathlib_Analysis_BoxIntegral_Box_Basic |
case coe.bot
ι : Type u_1
I J : Box ι
x y : ι → ℝ
a✝ : Box ι
⊢ ↑↑a✝ ⊆ ↑⊥ ↔ ↑a✝ ≤ ⊥ | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simp [subset_empty_iff] | @[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J := by
induction I using WithBot.recBotCoe; · simp
induction J using WithBot.recBotCoe; · | Mathlib.Analysis.BoxIntegral.Box.Basic.294_0.huZHUvFo9Qi2rb8 | @[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J | Mathlib_Analysis_BoxIntegral_Box_Basic |
case coe.coe
ι : Type u_1
I J : Box ι
x y : ι → ℝ
a✝¹ a✝ : Box ι
⊢ ↑↑a✝¹ ⊆ ↑↑a✝ ↔ ↑a✝¹ ≤ ↑a✝ | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simp [le_def] | @[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J := by
induction I using WithBot.recBotCoe; · simp
induction J using WithBot.recBotCoe; · simp [subset_empty_iff]
| Mathlib.Analysis.BoxIntegral.Box.Basic.294_0.huZHUvFo9Qi2rb8 | @[simp, norm_cast]
theorem withBotCoe_subset_iff {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) ⊆ J ↔ I ≤ J | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I✝ J✝ : Box ι
x y : ι → ℝ
I J : WithBot (Box ι)
⊢ ↑I = ↑J ↔ I = J | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simp only [Subset.antisymm_iff, ← le_antisymm_iff, withBotCoe_subset_iff] | @[simp, norm_cast]
theorem withBotCoe_inj {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) = J ↔ I = J := by
| Mathlib.Analysis.BoxIntegral.Box.Basic.301_0.huZHUvFo9Qi2rb8 | @[simp, norm_cast]
theorem withBotCoe_inj {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) = J ↔ I = J | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y l u : ι → ℝ
⊢ mk' l u = ⊥ ↔ ∃ i, u i ≤ l i | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | rw [mk'] | @[simp]
theorem mk'_eq_bot {l u : ι → ℝ} : mk' l u = ⊥ ↔ ∃ i, u i ≤ l i := by
| Mathlib.Analysis.BoxIntegral.Box.Basic.313_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_bot {l u : ι → ℝ} : mk' l u = ⊥ ↔ ∃ i, u i ≤ l i | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y l u : ι → ℝ
⊢ (if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) = ⊥ ↔ ∃ i, u i ≤ l i | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | split_ifs with h | @[simp]
theorem mk'_eq_bot {l u : ι → ℝ} : mk' l u = ⊥ ↔ ∃ i, u i ≤ l i := by
rw [mk']
| Mathlib.Analysis.BoxIntegral.Box.Basic.313_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_bot {l u : ι → ℝ} : mk' l u = ⊥ ↔ ∃ i, u i ≤ l i | Mathlib_Analysis_BoxIntegral_Box_Basic |
case pos
ι : Type u_1
I J : Box ι
x y l u : ι → ℝ
h : ∀ (i : ι), l i < u i
⊢ ↑{ lower := l, upper := u, lower_lt_upper := h } = ⊥ ↔ ∃ i, u i ≤ l i | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simpa using h | @[simp]
theorem mk'_eq_bot {l u : ι → ℝ} : mk' l u = ⊥ ↔ ∃ i, u i ≤ l i := by
rw [mk']
split_ifs with h <;> | Mathlib.Analysis.BoxIntegral.Box.Basic.313_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_bot {l u : ι → ℝ} : mk' l u = ⊥ ↔ ∃ i, u i ≤ l i | Mathlib_Analysis_BoxIntegral_Box_Basic |
case neg
ι : Type u_1
I J : Box ι
x y l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
⊢ ⊥ = ⊥ ↔ ∃ i, u i ≤ l i | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simpa using h | @[simp]
theorem mk'_eq_bot {l u : ι → ℝ} : mk' l u = ⊥ ↔ ∃ i, u i ≤ l i := by
rw [mk']
split_ifs with h <;> | Mathlib.Analysis.BoxIntegral.Box.Basic.313_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_bot {l u : ι → ℝ} : mk' l u = ⊥ ↔ ∃ i, u i ≤ l i | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y l u : ι → ℝ
⊢ mk' l u = ↑I ↔ l = I.lower ∧ u = I.upper | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | cases' I with lI uI hI | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper := by
| Mathlib.Analysis.BoxIntegral.Box.Basic.319_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper | Mathlib_Analysis_BoxIntegral_Box_Basic |
case mk
ι : Type u_1
J : Box ι
x y l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
⊢ mk' l u = ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | rw [mk'] | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper := by
cases' I with lI uI hI; | Mathlib.Analysis.BoxIntegral.Box.Basic.319_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper | Mathlib_Analysis_BoxIntegral_Box_Basic |
case mk
ι : Type u_1
J : Box ι
x y l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
⊢ (if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) =
↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
u = { lower... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | split_ifs with h | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper := by
cases' I with lI uI hI; rw [mk']; | Mathlib.Analysis.BoxIntegral.Box.Basic.319_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper | Mathlib_Analysis_BoxIntegral_Box_Basic |
case pos
ι : Type u_1
J : Box ι
x y l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
h : ∀ (i : ι), l i < u i
⊢ ↑{ lower := l, upper := u, lower_lt_upper := h } = ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
u = { lower := lI, upper := uI, l... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simp [WithBot.coe_eq_coe] | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper := by
cases' I with lI uI hI; rw [mk']; split_ifs with h
· | Mathlib.Analysis.BoxIntegral.Box.Basic.319_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper | Mathlib_Analysis_BoxIntegral_Box_Basic |
case neg
ι : Type u_1
J : Box ι
x y l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
h : ¬∀ (i : ι), l i < u i
⊢ ⊥ = ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | suffices l = lI → u ≠ uI by simpa | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper := by
cases' I with lI uI hI; rw [mk']; split_ifs with h
· simp [WithBot.coe_eq_coe]
· | Mathlib.Analysis.BoxIntegral.Box.Basic.319_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
J : Box ι
x y l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
h : ¬∀ (i : ι), l i < u i
this : l = lI → u ≠ uI
⊢ ⊥ = ↑{ lower := lI, upper := uI, lower_lt_upper := hI } ↔
l = { lower := lI, upper := uI, lower_lt_upper := hI }.lower ∧
u = { lower := lI, upper := uI, lower_lt_upper := hI }.upper | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | simpa | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper := by
cases' I with lI uI hI; rw [mk']; split_ifs with h
· simp [WithBot.coe_eq_coe]
· suffices l = lI → u ≠ uI by | Mathlib.Analysis.BoxIntegral.Box.Basic.319_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper | Mathlib_Analysis_BoxIntegral_Box_Basic |
case neg
ι : Type u_1
J : Box ι
x y l u lI uI : ι → ℝ
hI : ∀ (i : ι), lI i < uI i
h : ¬∀ (i : ι), l i < u i
⊢ l = lI → u ≠ uI | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | rintro rfl rfl | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper := by
cases' I with lI uI hI; rw [mk']; split_ifs with h
· simp [WithBot.coe_eq_coe]
· suffices l = lI → u ≠ uI by simpa
| Mathlib.Analysis.BoxIntegral.Box.Basic.319_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper | Mathlib_Analysis_BoxIntegral_Box_Basic |
case neg
ι : Type u_1
J : Box ι
x y l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
hI : ∀ (i : ι), l i < u i
⊢ False | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | exact h hI | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper := by
cases' I with lI uI hI; rw [mk']; split_ifs with h
· simp [WithBot.coe_eq_coe]
· suffices l = lI → u ≠ uI by simpa
rintro rfl rfl
| Mathlib.Analysis.BoxIntegral.Box.Basic.319_0.huZHUvFo9Qi2rb8 | @[simp]
theorem mk'_eq_coe {l u : ι → ℝ} : mk' l u = I ↔ l = I.lower ∧ u = I.upper | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y l u : ι → ℝ
⊢ ↑(mk' l u) = Set.pi univ fun i => Ioc (l i) (u i) | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | rw [mk'] | @[simp]
theorem coe_mk' (l u : ι → ℝ) : (mk' l u : Set (ι → ℝ)) = pi univ fun i ↦ Ioc (l i) (u i) := by
| Mathlib.Analysis.BoxIntegral.Box.Basic.328_0.huZHUvFo9Qi2rb8 | @[simp]
theorem coe_mk' (l u : ι → ℝ) : (mk' l u : Set (ι → ℝ)) = pi univ fun i ↦ Ioc (l i) (u i) | Mathlib_Analysis_BoxIntegral_Box_Basic |
ι : Type u_1
I J : Box ι
x y l u : ι → ℝ
⊢ ↑(if h : ∀ (i : ι), l i < u i then ↑{ lower := l, upper := u, lower_lt_upper := h } else ⊥) =
Set.pi univ fun i => Ioc (l i) (u i) | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Topology.Algebra.Order.MonotoneConvergence
import Mathlib.Topology.MetricSpace.Bounded
#align_import anal... | split_ifs with h | @[simp]
theorem coe_mk' (l u : ι → ℝ) : (mk' l u : Set (ι → ℝ)) = pi univ fun i ↦ Ioc (l i) (u i) := by
rw [mk']; | Mathlib.Analysis.BoxIntegral.Box.Basic.328_0.huZHUvFo9Qi2rb8 | @[simp]
theorem coe_mk' (l u : ι → ℝ) : (mk' l u : Set (ι → ℝ)) = pi univ fun i ↦ Ioc (l i) (u i) | Mathlib_Analysis_BoxIntegral_Box_Basic |
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