Theorem: CategoryTheory.Functor.preservesFiniteColimits_tfae
File: Mathlib/Algebra/Homology/ShortComplex/ExactFunctor.lean

Proof steps:
state_before: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
⊢ [∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F),
      Nonempty (PreservesFiniteColimits F)].TFAE
state_after: case tfae_1_to_2
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
⊢ (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)

C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
⊢ [∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F),
      Nonempty (PreservesFiniteColimits F)].TFAE
tactic: tfae_have 1 → 2
state_before: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
⊢ [∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F),
      Nonempty (PreservesFiniteColimits F)].TFAE
state_after: case tfae_2_to_3
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
⊢ (∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)

C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
tfae_2_to_3 :
  (∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)
⊢ [∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F),
      Nonempty (PreservesFiniteColimits F)].TFAE
tactic: tfae_have 2 → 3
state_before: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
tfae_2_to_3 :
  (∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)
⊢ [∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F),
      Nonempty (PreservesFiniteColimits F)].TFAE
state_after: case tfae_3_to_4
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
tfae_2_to_3 :
  (∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)
⊢ (∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)) → Nonempty (PreservesFiniteColimits F)

C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
tfae_2_to_3 :
  (∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)
tfae_3_to_4 :
  (∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)) → Nonempty (PreservesFiniteColimits F)
⊢ [∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F),
      Nonempty (PreservesFiniteColimits F)].TFAE
tactic: tfae_have 3 → 4
state_before: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
tfae_2_to_3 :
  (∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)
tfae_3_to_4 :
  (∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)) → Nonempty (PreservesFiniteColimits F)
⊢ [∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F),
      Nonempty (PreservesFiniteColimits F)].TFAE
state_after: case tfae_4_to_1
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
tfae_2_to_3 :
  (∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)
tfae_3_to_4 :
  (∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)) → Nonempty (PreservesFiniteColimits F)
⊢ Nonempty (PreservesFiniteColimits F) → ∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)

C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
tfae_2_to_3 :
  (∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)
tfae_3_to_4 :
  (∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)) → Nonempty (PreservesFiniteColimits F)
tfae_4_to_1 :
  Nonempty (PreservesFiniteColimits F) → ∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)
⊢ [∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F),
      Nonempty (PreservesFiniteColimits F)].TFAE
tactic: tfae_have 4 → 1
state_before: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
inst✝² : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)
tfae_2_to_3 :
  (∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g)) →
    ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)
tfae_3_to_4 :
  (∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F)) → Nonempty (PreservesFiniteColimits F)
tfae_4_to_1 :
  Nonempty (PreservesFiniteColimits F) → ∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g)
⊢ [∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ (S : ShortComplex C), S.Exact ∧ Epi S.g → (S.map F).Exact ∧ Epi (F.map S.g),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (PreservesColimit (parallelPair f 0) F),
      Nonempty (PreservesFiniteColimits F)].TFAE
state_after: no goals
tactic: tfae_finish