-- Mathlib/Algebra/Homology/ShortComplex/ExactFunctor.lean
/-- An additive functor which preserves homology preserves finite limits. -/
lemma preservesFiniteLimits_of_preservesHomology
    [HasFiniteProducts C] [HasKernels C] : PreservesFiniteLimits F

Proof state:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
this : HasBinaryBiproducts C
⊢ PreservesFiniteLimits F

Tactic:
have : HasEqualizers C := Preadditive.hasEqualizers_of_hasKernels

Result:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this✝¹ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
this✝ : HasBinaryBiproducts C
this : HasEqualizers C
⊢ PreservesFiniteLimits F

Proof state:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this✝² : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
this✝¹ : HasBinaryBiproducts C
this✝ : HasEqualizers C
this : HasZeroObject D
⊢ PreservesFiniteLimits F

Tactic:
exact preservesFiniteLimits_of_preservesKernels F

Result:
no goals

Proof state:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
⊢ PreservesFiniteLimits F

Tactic:
have : HasBinaryBiproducts C := HasBinaryBiproducts.of_hasBinaryProducts

Result:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
this : HasBinaryBiproducts C
⊢ PreservesFiniteLimits F

Proof state:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this✝¹ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
this✝ : HasBinaryBiproducts C
this : HasEqualizers C
⊢ PreservesFiniteLimits F

Tactic:
have : HasZeroObject D :=
  ⟨F.obj 0, by rw [IsZero.iff_id_eq_zero, ← F.map_id, id_zero, F.map_zero]⟩

Result:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this✝² : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
this✝¹ : HasBinaryBiproducts C
this✝ : HasEqualizers C
this : HasZeroObject D
⊢ PreservesFiniteLimits F

Proof state:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
⊢ PreservesFiniteLimits F

Tactic:
have := fun {X Y : C} (f : X ⟶ Y) ↦ PreservesHomology.preservesKernel F f

Result:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
⊢ PreservesFiniteLimits F

Proof state:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this✝¹ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
this✝ : HasBinaryBiproducts C
this : HasEqualizers C
⊢ IsZero (F.obj 0)

Tactic:
rw [IsZero.iff_id_eq_zero, ← F.map_id, id_zero, F.map_zero]

Result:
no goals