Theorem: CategoryTheory.GrothendieckTopology.Plus.toPlus_eq_mk
File: Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean

tactic: dsimp [mk, toPlus]
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ ((J.toPlus P).app { unop := X }) x = mk (Meq.mk ⊤ x)
state_after: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ (⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P X) { unop := ⊤ }) x =
    (colimit.ι (J.diagram P X) { unop := ⊤ }) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))

tactic: delta Cover.toMultiequalizer
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ (⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P X) { unop := ⊤ }) x =
    (colimit.ι (J.diagram P X) { unop := ⊤ }) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))
state_after: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ (Multiequalizer.lift (⊤.index P) (P.obj { unop := X }) (fun I => P.map I.f.op) ⋯ ≫
        colimit.ι (J.diagram P X) { unop := ⊤ })
      x =
    (colimit.ι (J.diagram P X) { unop := ⊤ }) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))

tactic: simp only [comp_apply]
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ (Multiequalizer.lift (⊤.index P) (P.obj { unop := X }) (fun I => P.map I.f.op) ⋯ ≫
        colimit.ι (J.diagram P X) { unop := ⊤ })
      x =
    (colimit.ι (J.diagram P X) { unop := ⊤ }) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))
state_after: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ (colimit.ι (J.diagram P X) { unop := ⊤ })
      ((Multiequalizer.lift (⊤.index P) (P.obj { unop := X }) (fun I => P.map I.f.op) ⋯) x) =
    (colimit.ι (J.diagram P X) { unop := ⊤ }) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))

tactic: apply congr_arg
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ (colimit.ι (J.diagram P X) { unop := ⊤ })
      ((Multiequalizer.lift (⊤.index P) (P.obj { unop := X }) (fun I => P.map I.f.op) ⋯) x) =
    (colimit.ι (J.diagram P X) { unop := ⊤ }) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))
state_after: case h
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ (Multiequalizer.lift (⊤.index P) (P.obj { unop := X }) (fun I => P.map I.f.op) ⋯) x =
    (Meq.equiv P ⊤).symm (Meq.mk ⊤ x)

tactic: apply (Meq.equiv P ⊤).injective
state_before: case h
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ (Multiequalizer.lift (⊤.index P) (P.obj { unop := X }) (fun I => P.map I.f.op) ⋯) x =
    (Meq.equiv P ⊤).symm (Meq.mk ⊤ x)
state_after: case h.a
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ (Meq.equiv P ⊤) ((Multiequalizer.lift (⊤.index P) (P.obj { unop := X }) (fun I => P.map I.f.op) ⋯) x) =
    (Meq.equiv P ⊤) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))

tactic: ext i
state_before: case h.a
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
⊢ (Meq.equiv P ⊤) ((Multiequalizer.lift (⊤.index P) (P.obj { unop := X }) (fun I => P.map I.f.op) ⋯) x) =
    (Meq.equiv P ⊤) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))
state_after: case h.a.h
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
i : ⊤.Arrow
⊢ ↑((Meq.equiv P ⊤) ((Multiequalizer.lift (⊤.index P) (P.obj { unop := X }) (fun I => P.map I.f.op) ⋯) x)) i =
    ↑((Meq.equiv P ⊤) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))) i

tactic: rw [Meq.equiv_apply, Equiv.apply_symm_apply, ← comp_apply, Multiequalizer.lift_ι]
state_before: case h.a.h
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
i : ⊤.Arrow
⊢ ↑((Meq.equiv P ⊤) ((Multiequalizer.lift (⊤.index P) (P.obj { unop := X }) (fun I => P.map I.f.op) ⋯) x)) i =
    ↑((Meq.equiv P ⊤) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))) i
state_after: case h.a.h
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
i : ⊤.Arrow
⊢ (P.map i.f.op) x = ↑(Meq.mk ⊤ x) i

tactic: rfl
state_before: case h.a.h
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj (P.obj { unop := X })
i : ⊤.Arrow
⊢ (P.map i.f.op) x = ↑(Meq.mk ⊤ x) i
state_after: no goals