state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
W✝ X✝ Y✝ Z✝ : Mat_ C
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.... | apply DMatrix.ext | instance : Category.{v₁} (Mat_ C) where
Hom := Hom
id := Hom.id
comp f g := f.comp g
id_comp f := by simp (config := { unfoldPartialApp := true }) [dite_comp]
comp_id f := by simp (config := { unfoldPartialApp := true }) [comp_dite]
assoc f g h := by
| Mathlib.CategoryTheory.Preadditive.Mat.106_0.xG9GKY7NTklnF73 | instance : Category.{v₁} (Mat_ C) where
Hom | Mathlib_CategoryTheory_Preadditive_Mat |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
W✝ X✝ Y✝ Z✝ : Mat_ C
f : W✝ ⟶ X✝
g : X✝ ⟶ Y✝
h : Y✝ ⟶ Z✝
⊢ ∀ (i : W✝.ι) (j : Z✝.ι), ((f ≫ g) ≫ h) i j = (f ≫ g ≫ h) 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.... | intros | instance : Category.{v₁} (Mat_ C) where
Hom := Hom
id := Hom.id
comp f g := f.comp g
id_comp f := by simp (config := { unfoldPartialApp := true }) [dite_comp]
comp_id f := by simp (config := { unfoldPartialApp := true }) [comp_dite]
assoc f g h := by
apply DMatrix.ext
| Mathlib.CategoryTheory.Preadditive.Mat.106_0.xG9GKY7NTklnF73 | instance : Category.{v₁} (Mat_ C) where
Hom | Mathlib_CategoryTheory_Preadditive_Mat |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
W✝ X✝ Y✝ Z✝ : Mat_ C
f : W✝ ⟶ X✝
g : X✝ ⟶ Y✝
h : Y✝ ⟶ Z✝
i✝ : W✝.ι
j✝ : Z✝.ι
⊢ ((f ≫ g) ≫ h) i✝ j✝ = (f ≫ g ≫ h) 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_rw [Hom.comp, sum_comp, comp_sum, Category.assoc] | instance : Category.{v₁} (Mat_ C) where
Hom := Hom
id := Hom.id
comp f g := f.comp g
id_comp f := by simp (config := { unfoldPartialApp := true }) [dite_comp]
comp_id f := by simp (config := { unfoldPartialApp := true }) [comp_dite]
assoc f g h := by
apply DMatrix.ext
intros
| Mathlib.CategoryTheory.Preadditive.Mat.106_0.xG9GKY7NTklnF73 | instance : Category.{v₁} (Mat_ C) where
Hom | Mathlib_CategoryTheory_Preadditive_Mat |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
W✝ X✝ Y✝ Z✝ : Mat_ C
f : W✝ ⟶ X✝
g : X✝ ⟶ Y✝
h : Y✝ ⟶ Z✝
i✝ : W✝.ι
j✝ : Z✝.ι
⊢ ∑ x : Y✝.ι, ∑ x_1 : X✝.ι, f i✝ x_1 ≫ g x_1 x ≫ h x j✝ = ∑ x : X✝.ι, ∑ j : Y✝.ι, f i✝ x ≫ g x j ≫ h j 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.... | rw [Finset.sum_comm] | instance : Category.{v₁} (Mat_ C) where
Hom := Hom
id := Hom.id
comp f g := f.comp g
id_comp f := by simp (config := { unfoldPartialApp := true }) [dite_comp]
comp_id f := by simp (config := { unfoldPartialApp := true }) [comp_dite]
assoc f g h := by
apply DMatrix.ext
intros
simp_rw [Hom.comp, s... | Mathlib.CategoryTheory.Preadditive.Mat.106_0.xG9GKY7NTklnF73 | instance : Category.{v₁} (Mat_ C) where
Hom | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i : M.ι
⊢ 𝟙 M i i = 𝟙 (X M 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.... | simp [id_apply] | @[simp]
theorem id_apply_self (M : Mat_ C) (i : M.ι) : (𝟙 M : Hom M M) i i = 𝟙 _ := by | Mathlib.CategoryTheory.Preadditive.Mat.136_0.xG9GKY7NTklnF73 | @[simp]
theorem id_apply_self (M : Mat_ C) (i : M.ι) : (𝟙 M : Hom M M) i i = 𝟙 _ | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
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_ C) (i j : M.ι) (h : i ≠ j) : (𝟙 M : Hom M M) i j = 0 := by
| Mathlib.CategoryTheory.Preadditive.Mat.141_0.xG9GKY7NTklnF73 | @[simp]
theorem id_apply_of_ne (M : Mat_ C) (i j : M.ι) (h : i ≠ j) : (𝟙 M : Hom M M) i j = 0 | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M N : Mat_ C
⊢ AddCommGroup (M ⟶ N) | /-
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 (DMatrix M.ι N.ι _) | instance (M N : Mat_ C) : AddCommGroup (M ⟶ N) := by
| Mathlib.CategoryTheory.Preadditive.Mat.167_0.xG9GKY7NTklnF73 | instance (M N : Mat_ C) : AddCommGroup (M ⟶ N) | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M N : Mat_ C
⊢ AddCommGroup (DMatrix M.ι N.ι fun i j => (fun i j => X M i ⟶ X N j) 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.... | infer_instance | instance (M N : Mat_ C) : AddCommGroup (M ⟶ N) := by
change AddCommGroup (DMatrix M.ι N.ι _)
| Mathlib.CategoryTheory.Preadditive.Mat.167_0.xG9GKY7NTklnF73 | instance (M N : Mat_ C) : AddCommGroup (M ⟶ N) | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M N K : Mat_ C
f f' : M ⟶ N
g : N ⟶ K
⊢ (f + f') ≫ g = f ≫ g + f' ≫ g | /-
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 : Preadditive (Mat_ C) where
add_comp M N K f f' g := by | Mathlib.CategoryTheory.Preadditive.Mat.177_0.xG9GKY7NTklnF73 | instance : Preadditive (Mat_ C) where
add_comp M N K f f' g | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M N K : Mat_ C
f f' : M ⟶ N
g : N ⟶ K
i✝ : M.ι
j✝ : K.ι
⊢ ((f + f') ≫ g) i✝ j✝ = (f ≫ g + f' ≫ g) 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 [Finset.sum_add_distrib] | instance : Preadditive (Mat_ C) where
add_comp M N K f f' g := by ext; | Mathlib.CategoryTheory.Preadditive.Mat.177_0.xG9GKY7NTklnF73 | instance : Preadditive (Mat_ C) where
add_comp M N K f f' g | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M N K : Mat_ C
f : M ⟶ N
g g' : N ⟶ K
⊢ f ≫ (g + g') = f ≫ g + f ≫ g' | /-
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 : Preadditive (Mat_ C) where
add_comp M N K f f' g := by ext; simp [Finset.sum_add_distrib]
comp_add M N K f g g' := by | Mathlib.CategoryTheory.Preadditive.Mat.177_0.xG9GKY7NTklnF73 | instance : Preadditive (Mat_ C) where
add_comp M N K f f' g | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M N K : Mat_ C
f : M ⟶ N
g g' : N ⟶ K
i✝ : M.ι
j✝ : K.ι
⊢ (f ≫ (g + g')) i✝ j✝ = (f ≫ g + f ≫ g') 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 [Finset.sum_add_distrib] | instance : Preadditive (Mat_ C) where
add_comp M N K f f' g := by ext; simp [Finset.sum_add_distrib]
comp_add M N K f g g' := by ext; | Mathlib.CategoryTheory.Preadditive.Mat.177_0.xG9GKY7NTklnF73 | instance : Preadditive (Mat_ C) where
add_comp M N K f f' g | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j : Fin n
x : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι
y : (f j).ι
⊢ (fun i j_1 => X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) i ⟶ X (f j) j_1) 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.... | refine' if h : x.1 = j then _ else 0 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j : Fin n
x : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι
y : (f j).ι
h : x.fst = j
⊢ (fun i j_1 => X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) i ⟶ X (f j) j_1) 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.... | refine' if h' : @Eq.ndrec (Fin n) x.1 (fun j => (f j).ι) x.2 _ h = y then _ else 0 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j : Fin n
x : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι
y : (f j).ι
h : x.fst = j
h' : h ▸ x.snd = y
⊢ (fun i j_1 => X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) i ⟶ X (f j) j_1) 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.... | apply eqToHom | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case p
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j : Fin n
x : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι
y : (f j).ι
h : x.fst = j
h' : h ▸ x.snd = y
⊢ X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) x = X (f j) 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.... | substs h h' | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case p
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
x : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι
⊢ X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) x = X (f x.fst) ((_ : x.fst = x.fst) ▸ x.snd) | /-
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 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j : Fin n
x : (f j).ι
y : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι
⊢ (fun i j_1 => X (f j) i ⟶ X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) j_1) 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.... | refine' if h : y.1 = j then _ else 0 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j : Fin n
x : (f j).ι
y : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι
h : y.fst = j
⊢ (fun i j_1 => X (f j) i ⟶ X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) j_1) 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.... | refine' if h' : @Eq.ndrec _ y.1 (fun j => (f j).ι) y.2 _ h = x then _ else 0 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j : Fin n
x : (f j).ι
y : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι
h : y.fst = j
h' : h ▸ y.snd = x
⊢ (fun i j_1 => X (f j) i ⟶ X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) j_1) 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.... | apply eqToHom | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case p
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j : Fin n
x : (f j).ι
y : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι
h : y.fst = j
h' : h ▸ y.snd = x
⊢ X (f j) x = X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) 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.... | substs h h' | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case p
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
y : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι
⊢ X (f y.fst) ((_ : y.fst = y.fst) ▸ y.snd) = X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) 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.... | rfl | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
⊢ (fun j x y =>
if h : y.fst = j then
if h' : h ▸ y.snd = x then
eqToHom (_ : X (f j) x = X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) y)
else 0
els... | /-
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 x y | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
x : (f j).ι
y : (f j').ι
⊢ ((fun j x y =>
if h : y.fst = j then
if h' : h ▸ y.snd = x then
eqToHom (_ : X (f j) x = X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.s... | /-
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 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
x : (f j).ι
y : (f j').ι
⊢ (∑ j_1 : (j : Fin n) × (f j).ι,
(if h : j_1.fst = j then if h' : h ▸ j_1.snd = x then eqToHom (_ : X (f j) x = X (f j_1.fst) j_1.snd) else 0
else 0) ≫
if h : j_1.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_rw [dite_comp, comp_dite] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
x : (f j).ι
y : (f j').ι
⊢ (∑ x_1 : (j : Fin n) × (f j).ι,
if h : x_1.fst = j then
if h_1 : (_ : x_1.fst = j) ▸ x_1.snd = x then
if h_2 : x_1.fst = j' then
if h_3 : (_ : x_1.fs... | /-
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 [ite_self, dite_eq_ite, dif_ctx_congr, Limits.comp_zero, Limits.zero_comp,
eqToHom_trans, Finset.sum_congr] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
x : (f j).ι
y : (f j').ι
⊢ (∑ x_1 : (j : Fin n) × (f j).ι,
if h : x_1.fst = j then
if h_1 : (_ : x_1.fst = j) ▸ x_1.snd = x then
if h_2 : x_1.fst = j' then
if h_3 : (_ : x_1.fs... | /-
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 [Finset.sum_sigma] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
x : (f j).ι
y : (f j').ι
⊢ (∑ a : Fin n,
∑ s in (fun x => Finset.univ) a,
if h : { fst := a, snd := s }.fst = j then
if h_1 : (_ : { fst := a, snd := s }.fst = j) ▸ { fst := a, snd := s }.... | /-
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 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
x : (f j).ι
y : (f j').ι
⊢ (∑ a : Fin n,
∑ s : (f a).ι,
if h : a = j then
if h_1 : (_ : a = j) ▸ s = x then
if h_2 : a = j' then if h_3 : (_ : a = j') ▸ s = y then 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 [if_congr, if_true, dif_ctx_congr, Finset.sum_dite_irrel, Finset.mem_univ,
Finset.sum_const_zero, Finset.sum_congr, Finset.sum_dite_eq'] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
x : (f j).ι
y : (f j').ι
⊢ (if h : j = j' then if h_1 : (_ : j = j') ▸ x = y then eqToHom (_ : X (f j) x = X (f j') y) else 0 else 0) =
dite (j = j') (fun h => eqToHom (_ : f j = f j')) (fun h => 0) 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.... | split_ifs with h h' | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
x : (f j).ι
y : (f j').ι
h : j = j'
h' : (_ : j = j') ▸ x = y
⊢ eqToHom (_ : X (f j) x = X (f j') y) = eqToHom (_ : f j = f j') 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.... | substs h h' | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j : Fin n
x : (f j).ι
⊢ eqToHom (_ : X (f j) x = X (f j) ((_ : j = j) ▸ x)) = eqToHom (_ : f j = f j) x ((_ : j = 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.... | simp only [CategoryTheory.eqToHom_refl, CategoryTheory.Mat_.id_apply_self] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
x : (f j).ι
y : (f j').ι
h : j = j'
h' : ¬(_ : j = j') ▸ x = y
⊢ 0 = eqToHom (_ : f j = f j') 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.... | subst h | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j : Fin n
x y : (f j).ι
h' : ¬(_ : j = j) ▸ x = y
⊢ 0 = eqToHom (_ : f j = f j) 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.... | rw [eqToHom_refl, id_apply_of_ne _ _ _ h'] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
j j' : Fin n
x : (f j).ι
y : (f j').ι
h : ¬j = j'
⊢ 0 = OfNat.ofNat 0 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.... | rfl | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
⊢ ∑ j : Fin n,
Bicone.π
(Bicone.mk (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd)
(fun j x y =>
if h : x.fst = j then
if h' : h ▸ x.snd = y then
eq... | /-
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 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
⊢ (∑ j : Fin n,
(fun x y =>
if h : x.fst = j then if h' : h ▸ x.snd = y then eqToHom (_ : X (f x.fst) x.snd = X (f j) y) else 0 else 0) ≫
fun x y =>
if h : y.fst = j then if h' : h ▸ y.snd = x then eq... | /-
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.... | ext1 ⟨i, j⟩ | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
⊢ ∀ (j_1 : (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd).ι),
Finset.sum Finset.univ
(fun j =>
(fun x y =>
if h : x.fst = j then if h' : h ▸ x.snd = y then... | /-
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 ⟨i', j'⟩ | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ Finset.sum Finset.univ
(fun j =>
(fun x y =>
if h : x.fst = j then if h' : h ▸ x.snd = y then eqToHom (_ : X (f x.fst) x.snd = X (f j) y) else 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.... | rw [Finset.sum_apply, Finset.sum_apply] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ ∑ c : Fin n,
((fun x y =>
if h : x.fst = c then if h' : h ▸ x.snd = y then eqToHom (_ : X (f x.fst) x.snd = X (f c) y) else 0
else 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.... | dsimp | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ (∑ c : Fin n,
∑ j_1 : (f c).ι,
(if h : i = c then if h' : h ▸ j = j_1 then eqToHom (_ : X (f i) j = X (f c) j_1) else 0 else 0) ≫
if h : i' = c th... | /-
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 [Finset.sum_eq_single i] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ (∑ j_1 : (f i).ι,
(if h : i = i then if h' : h ▸ j = j_1 then eqToHom (_ : X (f i) j = X (f i) j_1) else 0 else 0) ≫
if h : i' = i then if h' : h ▸ j' = 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.... | rotate_left | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ ∀ b ∈ Finset.univ,
b ≠ i →
(∑ j_1 : (f b).ι,
(if h : i = b then if h' : h ▸ j = j_1 then eqToHom (_ : X (f i) j = X (f b) j_1) else 0 else 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.... | intro b _ hb | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
b : Fin n
a✝ : b ∈ Finset.univ
hb : b ≠ i
⊢ (∑ j_1 : (f b).ι,
(if h : i = b then if h' : h ▸ j = j_1 then eqToHom (_ : X (f i) j = X (f b) j_1) else 0 else 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.... | apply Finset.sum_eq_zero | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₀.h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
b : Fin n
a✝ : b ∈ Finset.univ
hb : b ≠ i
⊢ ∀ x ∈ Finset.univ,
((if h : i = b then if h' : h ▸ j = x then eqToHom (_ : X (f i) j = X (f b) x) else 0 else 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.... | intro x _ | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₀.h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
b : Fin n
a✝¹ : b ∈ Finset.univ
hb : b ≠ i
x : (f b).ι
a✝ : x ∈ Finset.univ
⊢ ((if h : i = b then if h' : h ▸ j = x then eqToHom (_ : X (f i) j = X (f b) x) else 0 els... | /-
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 [dif_neg hb.symm, zero_comp] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₁
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ i ∉ Finset.univ →
(∑ j_1 : (f i).ι,
(if h : i = i then if h' : h ▸ j = j_1 then eqToHom (_ : X (f i) j = X (f i) j_1) else 0 else 0) ≫
if h : 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.... | intro hi | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₁
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
hi : i ∉ Finset.univ
⊢ (∑ j_1 : (f i).ι,
(if h : i = i then if h' : h ▸ j = j_1 then eqToHom (_ : X (f i) j = X (f i) j_1) else 0 else 0) ≫
if h : i' = i t... | /-
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 at hi | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ (∑ j_1 : (f i).ι,
(if h : i = i then if h' : h ▸ j = j_1 then eqToHom (_ : X (f i) j = X (f i) j_1) else 0 else 0) ≫
if h : i' = i then if h' : h ▸ j' = 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.... | rw [Finset.sum_eq_single j] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ ((if h : i = i then if h' : h ▸ j = j then eqToHom (_ : X (f i) j = X (f i) j) else 0 else 0) ≫
if h : i' = i then if h' : h ▸ j' = j then eqToHom (_ : X (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.... | rotate_left | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ ∀ b ∈ Finset.univ,
b ≠ j →
((if h : i = i then if h' : h ▸ j = b then eqToHom (_ : X (f i) j = X (f i) b) else 0 else 0) ≫
if h : i' = i then if 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.... | intro b _ hb | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
b : (f i).ι
a✝ : b ∈ Finset.univ
hb : b ≠ j
⊢ ((if h : i = i then if h' : h ▸ j = b then eqToHom (_ : X (f i) j = X (f i) b) else 0 else 0) ≫
if h : i' = i then 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.... | rw [dif_pos rfl, dif_neg, zero_comp] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₀.hnc
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
b : (f i).ι
a✝ : b ∈ Finset.univ
hb : b ≠ j
⊢ ¬(_ : i = i) ▸ j = b | /-
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 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₀.hnc
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
b : (f i).ι
a✝ : b ∈ Finset.univ
hb : b ≠ j
⊢ ¬j = b | /-
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.... | tauto | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₁
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ j ∉ Finset.univ →
((if h : i = i then if h' : h ▸ j = j then eqToHom (_ : X (f i) j = X (f i) j) else 0 else 0) ≫
if h : i' = i then if h' : h ▸ j' = j the... | /-
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.... | intro hj | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk.h₁
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
hj : j ∉ Finset.univ
⊢ ((if h : i = i then if h' : h ▸ j = j then eqToHom (_ : X (f i) j = X (f i) j) else 0 else 0) ≫
if h : i' = i then if h' : h ▸ j' = j then 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 at hj | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ ((if h : i = i then if h' : h ▸ j = j then eqToHom (_ : X (f i) j = X (f i) j) else 0 else 0) ≫
if h : i' = i then if h' : h ▸ j' = j then eqToHom (_ : X (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 [eqToHom_refl, dite_eq_ite, ite_true, Category.id_comp, ne_eq,
Sigma.mk.inj_iff, not_and, id_def] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ (if h : i' = i then if h' : h ▸ j' = j then eqToHom (_ : X (f i) j = X (f i') j') else 0 else 0) =
if h : i = i' ∧ HEq j j' then
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.... | by_cases h : i' = i | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
h : i' = i
⊢ (if h : i' = i then if h' : h ▸ j' = j then eqToHom (_ : X (f i) j = X (f i') j') else 0 else 0) =
if h : i = i' ∧ HEq j j' then
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.... | subst h | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i' : Fin n
j' j : (f i').ι
⊢ (if h : i' = i' then if h' : h ▸ j' = j then eqToHom (_ : X (f i') j = X (f i') j') else 0 else 0) =
if h : i' = i' ∧ HEq j j' then
eqToHom
(_ :
X (mk ((j : Fin n) × ... | /-
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 [dif_pos rfl] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i' : Fin n
j' j : (f i').ι
⊢ (if h' : (_ : i' = i') ▸ j' = j then eqToHom (_ : X (f i') j = X (f i') j') else 0) =
if h : i' = i' ∧ HEq j j' then
eqToHom
(_ :
X (mk ((j : Fin n) × (f j).ι) fun p ... | /-
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 [heq_eq_eq, true_and] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i' : Fin n
j' j : (f i').ι
⊢ (if h' : j' = j then eqToHom (_ : X (f i') j = X (f i') j') else 0) =
if h : j = j' then
eqToHom
(_ :
X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) { fst ... | /-
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' = j | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i' : Fin n
j' j : (f i').ι
h : j' = j
⊢ (if h' : j' = j then eqToHom (_ : X (f i') j = X (f i') j') else 0) =
if h : j = j' then
eqToHom
(_ :
X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.... | /-
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 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i' : Fin n
j' : (f i').ι
⊢ (if h' : j' = j' then eqToHom (_ : X (f i') j' = X (f i') j') else 0) =
if h : j' = j' then
eqToHom
(_ :
X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p.snd) { fst... | /-
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 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i' : Fin n
j' j : (f i').ι
h : ¬j' = j
⊢ (if h' : j' = j then eqToHom (_ : X (f i') j = X (f i') j') else 0) =
if h : j = j' then
eqToHom
(_ :
X (mk ((j : Fin n) × (f j).ι) fun p => X (f p.fst) p... | /-
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 [dif_neg h, dif_neg (Ne.symm h)] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
h : ¬i' = i
⊢ (if h : i' = i then if h' : h ▸ j' = j then eqToHom (_ : X (f i) j = X (f i') j') else 0 else 0) =
if h : i = i' ∧ HEq j j' then
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.... | rw [dif_neg h, dif_neg] | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
case neg.hnc
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
h : ¬i' = i
⊢ ¬(i = i' ∧ HEq j 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.... | tauto | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib.CategoryTheory.Preadditive.Mat.183_0.xG9GKY7NTklnF73 | /-- We now prove that `Mat_ C` has finite biproducts.
Be warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,
and so the internal indexing of a biproduct may have nothing to do with the external indexing,
even though the construction we give uses a sigma type.
See however `isoBiproductEmbedding`.
-/
instan... | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : Preadditive C
D : Type u_1
inst✝¹ : Category.{v₁, u_1} D
inst✝ : Preadditive D
M : Mat_ C
⊢ (mapMat_ (𝟭 C)).obj M = (𝟭 (Mat_ C)).obj 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.... | cases M | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) :=
NatIso.ofComponents (fun M => eqToIso (by | Mathlib.CategoryTheory.Preadditive.Mat.276_0.xG9GKY7NTklnF73 | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) | Mathlib_CategoryTheory_Preadditive_Mat |
case mk
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : Preadditive C
D : Type u_1
inst✝¹ : Category.{v₁, u_1} D
inst✝ : Preadditive D
ι✝ : Type
fintype✝ : Fintype ι✝
X✝ : ι✝ → C
⊢ (mapMat_ (𝟭 C)).obj (Mat_.mk ι✝ X✝) = (𝟭 (Mat_ C)).obj (Mat_.mk ι✝ 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.... | rfl | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) :=
NatIso.ofComponents (fun M => eqToIso (by cases M; | Mathlib.CategoryTheory.Preadditive.Mat.276_0.xG9GKY7NTklnF73 | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : Preadditive C
D : Type u_1
inst✝¹ : Category.{v₁, u_1} D
inst✝ : Preadditive D
M N : Mat_ C
f : M ⟶ N
⊢ (mapMat_ (𝟭 C)).map f ≫ ((fun M => eqToIso (_ : (mapMat_ (𝟭 C)).obj M = (𝟭 (Mat_ C)).obj M)) N).hom =
((fun M => eqToIso (_ : (mapMat_ (𝟭 C)).obj M = (𝟭 (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.... | ext | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) :=
NatIso.ofComponents (fun M => eqToIso (by cases M; rfl)) fun {M N} f => by
| Mathlib.CategoryTheory.Preadditive.Mat.276_0.xG9GKY7NTklnF73 | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : Preadditive C
D : Type u_1
inst✝¹ : Category.{v₁, u_1} D
inst✝ : Preadditive D
M N : Mat_ C
f : M ⟶ N
i✝ : ((mapMat_ (𝟭 C)).obj M).ι
j✝ : ((𝟭 (Mat_ C)).obj N).ι
⊢ ((mapMat_ (𝟭 C)).map f ≫ ((fun M => eqToIso (_ : (mapMat_ (𝟭 C)).obj M = (𝟭 (Mat_ C)).obj M)) N... | /-
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 M | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) :=
NatIso.ofComponents (fun M => eqToIso (by cases M; rfl)) fun {M N} f => by
ext
| Mathlib.CategoryTheory.Preadditive.Mat.276_0.xG9GKY7NTklnF73 | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : Preadditive C
D : Type u_1
inst✝¹ : Category.{v₁, u_1} D
inst✝ : Preadditive D
N : Mat_ C
j✝ : ((𝟭 (Mat_ C)).obj N).ι
ι✝ : Type
fintype✝ : Fintype ι✝
X✝ : ι✝ → C
f : Mat_.mk ι✝ X✝ ⟶ N
i✝ : ((mapMat_ (𝟭 C)).obj (Mat_.mk ι✝ X✝)).ι
⊢ ((mapMat_ (𝟭 C)).map 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.... | cases N | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) :=
NatIso.ofComponents (fun M => eqToIso (by cases M; rfl)) fun {M N} f => by
ext
cases M; | Mathlib.CategoryTheory.Preadditive.Mat.276_0.xG9GKY7NTklnF73 | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : Preadditive C
D : Type u_1
inst✝¹ : Category.{v₁, u_1} D
inst✝ : Preadditive D
ι✝¹ : Type
fintype✝¹ : Fintype ι✝¹
X✝¹ : ι✝¹ → C
i✝ : ((mapMat_ (𝟭 C)).obj (Mat_.mk ι✝¹ X✝¹)).ι
ι✝ : Type
fintype✝ : Fintype ι✝
X✝ : ι✝ → C
j✝ : ((𝟭 (Mat_ C)).obj (Mat_.mk ι✝ 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 [comp_dite, dite_comp] | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) :=
NatIso.ofComponents (fun M => eqToIso (by cases M; rfl)) fun {M N} f => by
ext
cases M; cases N
| Mathlib.CategoryTheory.Preadditive.Mat.276_0.xG9GKY7NTklnF73 | /-- The identity functor induces the identity functor on matrix categories.
-/
@[simps!]
def mapMatId : (𝟭 C).mapMat_ ≅ 𝟭 (Mat_ C) | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : Preadditive C
D : Type u_1
inst✝⁵ : Category.{v₁, u_1} D
inst✝⁴ : Preadditive D
E : Type u_2
inst✝³ : Category.{v₁, u_2} E
inst✝² : Preadditive E
F : C ⥤ D
inst✝¹ : Additive F
G : D ⥤ E
inst✝ : Additive G
M : Mat_ C
⊢ (mapMat_ (F ⋙ G)).obj M = (mapMat_ F ⋙ mapMat_ G).ob... | /-
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 M | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ :=
NatIso.ofComponents (fun M => eqToIso (by | Mathlib.CategoryTheory.Preadditive.Mat.287_0.xG9GKY7NTklnF73 | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ | Mathlib_CategoryTheory_Preadditive_Mat |
case mk
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : Preadditive C
D : Type u_1
inst✝⁵ : Category.{v₁, u_1} D
inst✝⁴ : Preadditive D
E : Type u_2
inst✝³ : Category.{v₁, u_2} E
inst✝² : Preadditive E
F : C ⥤ D
inst✝¹ : Additive F
G : D ⥤ E
inst✝ : Additive G
ι✝ : Type
fintype✝ : Fintype ι✝
X✝ : ι✝ → C
⊢ (mapMat_ (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.... | rfl | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ :=
NatIso.ofComponents (fun M => eqToIso (by cases M; | Mathlib.CategoryTheory.Preadditive.Mat.287_0.xG9GKY7NTklnF73 | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : Preadditive C
D : Type u_1
inst✝⁵ : Category.{v₁, u_1} D
inst✝⁴ : Preadditive D
E : Type u_2
inst✝³ : Category.{v₁, u_2} E
inst✝² : Preadditive E
F : C ⥤ D
inst✝¹ : Additive F
G : D ⥤ E
inst✝ : Additive G
M N : Mat_ C
f : M ⟶ N
⊢ (mapMat_ (F ⋙ G)).map f ≫ ((fun M => eqT... | /-
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 | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ :=
NatIso.ofComponents (fun M => eqToIso (by cases M; rfl)) f... | Mathlib.CategoryTheory.Preadditive.Mat.287_0.xG9GKY7NTklnF73 | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ | Mathlib_CategoryTheory_Preadditive_Mat |
case H
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : Preadditive C
D : Type u_1
inst✝⁵ : Category.{v₁, u_1} D
inst✝⁴ : Preadditive D
E : Type u_2
inst✝³ : Category.{v₁, u_2} E
inst✝² : Preadditive E
F : C ⥤ D
inst✝¹ : Additive F
G : D ⥤ E
inst✝ : Additive G
M N : Mat_ C
f : M ⟶ N
i✝ : ((mapMat_ (F ⋙ G)).obj M).ι
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.... | cases M | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ :=
NatIso.ofComponents (fun M => eqToIso (by cases M; rfl)) f... | Mathlib.CategoryTheory.Preadditive.Mat.287_0.xG9GKY7NTklnF73 | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : Preadditive C
D : Type u_1
inst✝⁵ : Category.{v₁, u_1} D
inst✝⁴ : Preadditive D
E : Type u_2
inst✝³ : Category.{v₁, u_2} E
inst✝² : Preadditive E
F : C ⥤ D
inst✝¹ : Additive F
G : D ⥤ E
inst✝ : Additive G
N : Mat_ C
j✝ : ((mapMat_ F ⋙ mapMat_ G).obj N).ι
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.... | cases N | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ :=
NatIso.ofComponents (fun M => eqToIso (by cases M; rfl)) f... | Mathlib.CategoryTheory.Preadditive.Mat.287_0.xG9GKY7NTklnF73 | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ | Mathlib_CategoryTheory_Preadditive_Mat |
case H.mk.mk
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : Preadditive C
D : Type u_1
inst✝⁵ : Category.{v₁, u_1} D
inst✝⁴ : Preadditive D
E : Type u_2
inst✝³ : Category.{v₁, u_2} E
inst✝² : Preadditive E
F : C ⥤ D
inst✝¹ : Additive F
G : D ⥤ E
inst✝ : Additive G
ι✝¹ : Type
fintype✝¹ : Fintype ι✝¹
X✝¹ : ι✝¹ → C
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.... | simp [comp_dite, dite_comp] | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ :=
NatIso.ofComponents (fun M => eqToIso (by cases M; rfl)) f... | Mathlib.CategoryTheory.Preadditive.Mat.287_0.xG9GKY7NTklnF73 | /-- Composite functors induce composite functors on matrix categories.
-/
@[simps!]
def mapMatComp {E : Type*} [Category.{v₁} E] [Preadditive E] (F : C ⥤ D) [Functor.Additive F]
(G : D ⥤ E) [Functor.Additive G] : (F ⋙ G).mapMat_ ≅ F.mapMat_ ⋙ G.mapMat_ | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
x✝ : C
⊢ { obj := fun X => mk PUnit.{1} fun x => X, map := fun {X Y} f x x => f }.map (𝟙 x✝) =
𝟙 ({ obj := fun X => mk PUnit.{1} fun x => X, map := fun {X Y} f x x => f }.obj 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.... | ext ⟨⟩ | /-- The embedding of `C` into `Mat_ C` as one-by-one matrices.
(We index the summands by `punit`.) -/
@[simps]
def embedding : C ⥤ Mat_ C where
obj X := ⟨PUnit, fun _ => X⟩
map f _ _ := f
map_id _ := by | Mathlib.CategoryTheory.Preadditive.Mat.303_0.xG9GKY7NTklnF73 | /-- The embedding of `C` into `Mat_ C` as one-by-one matrices.
(We index the summands by `punit`.) -/
@[simps]
def embedding : C ⥤ Mat_ C where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case H.unit
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
x✝ : C
j✝ : ({ obj := fun X => mk PUnit.{1} fun x => X, map := fun {X Y} f x x => f }.obj x✝).ι
⊢ { obj := fun X => mk PUnit.{1} fun x => X, map := fun {X Y} f x x => f }.map (𝟙 x✝) PUnit.unit j✝ =
𝟙 ({ obj := fun X => mk PUnit.{1} fun 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 | /-- The embedding of `C` into `Mat_ C` as one-by-one matrices.
(We index the summands by `punit`.) -/
@[simps]
def embedding : C ⥤ Mat_ C where
obj X := ⟨PUnit, fun _ => X⟩
map f _ _ := f
map_id _ := by ext ⟨⟩; | Mathlib.CategoryTheory.Preadditive.Mat.303_0.xG9GKY7NTklnF73 | /-- The embedding of `C` into `Mat_ C` as one-by-one matrices.
(We index the summands by `punit`.) -/
@[simps]
def embedding : C ⥤ Mat_ C where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
X✝ Y✝ Z✝ : C
x✝¹ : X✝ ⟶ Y✝
x✝ : Y✝ ⟶ Z✝
⊢ { obj := fun X => mk PUnit.{1} fun x => X, map := fun {X Y} f x x => f }.map (x✝¹ ≫ x✝) =
{ obj := fun X => mk PUnit.{1} fun x => X, map := fun {X Y} f x x => f }.map x✝¹ ≫
{ obj := fun X => mk PUnit.{1} 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.... | ext ⟨⟩ | /-- The embedding of `C` into `Mat_ C` as one-by-one matrices.
(We index the summands by `punit`.) -/
@[simps]
def embedding : C ⥤ Mat_ C where
obj X := ⟨PUnit, fun _ => X⟩
map f _ _ := f
map_id _ := by ext ⟨⟩; simp
map_comp _ _ := by | Mathlib.CategoryTheory.Preadditive.Mat.303_0.xG9GKY7NTklnF73 | /-- The embedding of `C` into `Mat_ C` as one-by-one matrices.
(We index the summands by `punit`.) -/
@[simps]
def embedding : C ⥤ Mat_ C where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
case H.unit
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
X✝ Y✝ Z✝ : C
x✝¹ : X✝ ⟶ Y✝
x✝ : Y✝ ⟶ Z✝
j✝ : ({ obj := fun X => mk PUnit.{1} fun x => X, map := fun {X Y} f x x => f }.obj Z✝).ι
⊢ { obj := fun X => mk PUnit.{1} fun x => X, map := fun {X Y} f x x => f }.map (x✝¹ ≫ x✝) PUnit.unit 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.... | simp | /-- The embedding of `C` into `Mat_ C` as one-by-one matrices.
(We index the summands by `punit`.) -/
@[simps]
def embedding : C ⥤ Mat_ C where
obj X := ⟨PUnit, fun _ => X⟩
map f _ _ := f
map_id _ := by ext ⟨⟩; simp
map_comp _ _ := by ext ⟨⟩; | Mathlib.CategoryTheory.Preadditive.Mat.303_0.xG9GKY7NTklnF73 | /-- The embedding of `C` into `Mat_ C` as one-by-one matrices.
(We index the summands by `punit`.) -/
@[simps]
def embedding : C ⥤ Mat_ C where
obj X | Mathlib_CategoryTheory_Preadditive_Mat |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
⊢ ((biproduct.lift fun i j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) ≫
biproduct.desc fun i j k => if h : i = k then eqToHom (_ : X M i = X M k) else 0) =
𝟙 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.... | simp only [biproduct.lift_desc] | /-- 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✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
⊢ (∑ j : M.ι,
(fun j_1 k => if h : j_1 = j then eqToHom (_ : X M j_1 = X M j) else 0) ≫ fun j_1 k =>
if h : j = k then eqToHom (_ : X M j = X M k) else 0) =
𝟙 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.... | funext i j | /-- 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 h.h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
⊢ Finset.sum Finset.univ
(fun j =>
(fun j_1 k => if h : j_1 = j then eqToHom (_ : X M j_1 = X M j) else 0) ≫ fun j_1 k =>
if h : j = k then eqToHom (_ : X M j = X M k) else 0)
i j =
𝟙 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.... | dsimp [id_def] | /-- 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 h.h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
⊢ Finset.sum Finset.univ
(fun j =>
(fun j_1 k => if h : j_1 = j then eqToHom (_ : X M j_1 = X M j) else 0) ≫ fun j_1 k =>
if h : j = k then eqToHom (_ : X M j = X M k) else 0)
i j =
if h : 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.... | rw [Finset.sum_apply, Finset.sum_apply, Finset.sum_eq_single i] | /-- 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 h.h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
⊢ ((fun j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) ≫ fun j k =>
if h : i = k then eqToHom (_ : X M i = X M k) else 0)
i j =
if h : i = j then eqToHom (_ : X M i = X M j) else 0
case h.h.h₀
C ... | /-
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.... | rotate_left | /-- 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 h.h.h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
⊢ ∀ b ∈ Finset.univ,
b ≠ i →
((fun j k => if h : j = b then eqToHom (_ : X M j = X M b) else 0) ≫ fun j k =>
if h : b = k then eqToHom (_ : X M b = X M k) else 0)
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.... | intro b _ hb | /-- 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 h.h.h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j b : M.ι
a✝ : b ∈ Finset.univ
hb : b ≠ i
⊢ ((fun j k => if h : j = b then eqToHom (_ : X M j = X M b) else 0) ≫ fun j k =>
if h : b = k then eqToHom (_ : X M b = X M k) else 0)
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.... | dsimp | /-- 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 h.h.h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j b : M.ι
a✝ : b ∈ Finset.univ
hb : b ≠ i
⊢ (∑ j_1 in {PUnit.unit},
(if h : i = b then eqToHom (_ : X M i = X M b) else 0) ≫ if h : b = j then eqToHom (_ : X M b = X M j) else 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.... | simp only [Finset.sum_const, Finset.card_singleton, one_smul] | /-- 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 h.h.h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j b : M.ι
a✝ : b ∈ Finset.univ
hb : b ≠ i
⊢ ((if h : i = b then eqToHom (_ : X M i = X M b) else 0) ≫ if h : b = j then eqToHom (_ : X M b = X M j) else 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.... | rw [dif_neg hb.symm, zero_comp] | /-- 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 h.h.h₁
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
⊢ i ∉ Finset.univ →
((fun j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) ≫ fun j k =>
if h : i = k then eqToHom (_ : X M i = X M k) else 0)
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.... | intro 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 h.h.h₁
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
h : i ∉ Finset.univ
⊢ ((fun j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) ≫ fun j k =>
if h : i = k then eqToHom (_ : X M i = X M k) else 0)
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 at 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 h.h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
⊢ ((fun j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) ≫ fun j k =>
if h : i = k then eqToHom (_ : X M i = X M k) else 0)
i j =
if h : i = j then eqToHom (_ : X M i = X M j) else 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 | /-- 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✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
⊢ ((biproduct.desc fun i j k => if h : i = k then eqToHom (_ : X M i = X M k) else 0) ≫
biproduct.lift fun i j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) =
𝟙 (⨁ fun i => (embedding C).obj (X M 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.... | apply biproduct.hom_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
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
⊢ ∀ (j : M.ι),
((biproduct.desc fun i j k => if h : i = k then eqToHom (_ : X M i = X M k) else 0) ≫
biproduct.lift fun i j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) ≫
biproduct.π (fun i => (embedding... | /-
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.... | intro i | /-- 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
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i : M.ι
⊢ ((biproduct.desc fun i j k => if h : i = k then eqToHom (_ : X M i = X M k) else 0) ≫
biproduct.lift fun i j k => if h : j = i then eqToHom (_ : X M j = X M i) else 0) ≫
biproduct.π (fun i => (embedding C).obj (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.... | apply biproduct.hom_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
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i : M.ι
⊢ ∀ (j : M.ι),
biproduct.ι (fun i => (embedding C).obj (X M i)) j ≫
((biproduct.desc fun i j k => if h : i = k then eqToHom (_ : X M i = X M k) else 0) ≫
biproduct.lift fun i j k => if h : j = i then eqToH... | /-
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.... | intro j | /-- 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
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j : M.ι
⊢ biproduct.ι (fun i => (embedding C).obj (X M i)) j ≫
((biproduct.desc fun i j k => if h : i = k then eqToHom (_ : X M i = X M k) else 0) ≫
biproduct.lift fun i j k => if h : j = i 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.... | simp only [Category.id_comp, Category.assoc, biproduct.lift_π, biproduct.ι_desc_assoc,
biproduct.ι_π] | /-- 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 |
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