Theorem: Profinite.NobelingProof.succ_exact
File: Mathlib/Topology/Category/Profinite/Nobeling.lean

Proof steps:
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
⊢ ModuleCat.ofHom (πs C o) ≫ ModuleCat.ofHom (Linear_CC' C hsC ho) = 0
state_after: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
x✝ : ↑(ModuleCat.of ℤ (LocallyConstant ↑(π C fun x => ord I x < o) ℤ))
⊢ (ModuleCat.ofHom (πs C o) ≫ ModuleCat.ofHom (Linear_CC' C hsC ho)) x✝ = 0 x✝
tactic: ext
state_before: case h
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
x✝ : ↑(ModuleCat.of ℤ (LocallyConstant ↑(π C fun x => ord I x < o) ℤ))
⊢ (ModuleCat.ofHom (πs C o) ≫ ModuleCat.ofHom (Linear_CC' C hsC ho)) x✝ = 0 x✝
state_after: no goals
tactic: apply CC_comp_zero
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
⊢ (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).Exact
state_after: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
⊢ ∀ (x₂ : ↑(ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).X₂),
    (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).g x₂ = 0 →
      ∃ x₁, (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).f x₁ = x₂
tactic: rw [ShortComplex.moduleCat_exact_iff]
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
⊢ ∀ (x₂ : ↑(ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).X₂),
    (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).g x₂ = 0 →
      ∃ x₁, (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).f x₁ = x₂
state_after: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : ↑(ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).X₂
⊢ (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).g f = 0 →
    ∃ x₁, (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).f x₁ = f
tactic: intro f
state_before: I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : ↑(ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).X₂
⊢ (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).g f = 0 →
    ∃ x₁, (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).f x₁ = f
state_after: no goals
tactic: exact CC_exact C hC hsC ho