Theorem: AlgebraicGeometry.PresheafedSpace.sheafIsoOfIso_hom
File: Mathlib/Geometry/RingedSpace/PresheafedSpace.lean

Proof steps:
state_before: C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
⊢ H.hom.c ≫ Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c = 𝟙 Y.presheaf
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ (H.hom.c ≫ Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c).app (op U) = (𝟙 Y.presheaf).app (op U)
tactic: ext U
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ (H.hom.c ≫ Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c).app (op U) = (𝟙 Y.presheaf).app (op U)
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ H.hom.c.app (op U) ≫ (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c).app (op U) =
    (𝟙 Y.presheaf).app (op U)
tactic: rw [NatTrans.comp_app]
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ H.hom.c.app (op U) ≫ (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c).app (op U) =
    (𝟙 Y.presheaf).app (op U)
state_after: no goals
tactic: simpa using congr_arg (fun f => f ≫ eqToHom _) (congr_app H.inv_hom_id (op U))
state_before: C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
⊢ Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c ≫ H.hom.c =
    𝟙 ((Presheaf.pushforward C H.hom.base).obj X.presheaf)
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c ≫ H.hom.c).app (op U) =
    (𝟙 ((Presheaf.pushforward C H.hom.base).obj X.presheaf)).app (op U)
tactic: ext U
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c ≫ H.hom.c).app (op U) =
    (𝟙 ((Presheaf.pushforward C H.hom.base).obj X.presheaf)).app (op U)
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c).app (op U) ≫ H.hom.c.app (op U) =
    (𝟙 ((Presheaf.pushforward C H.hom.base).obj X.presheaf)).app (op U)
tactic: dsimp
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c).app (op U) ≫ H.hom.c.app (op U) =
    (𝟙 ((Presheaf.pushforward C H.hom.base).obj X.presheaf)).app (op U)
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c).app (op U) ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
tactic: rw [NatTrans.id_app]
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c).app (op U) ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
tactic: simp only [Presheaf.pushforwardToOfIso_app, Iso.symm_inv, mapIso_hom, forget_map,
      Iso.symm_hom, mapIso_inv, eqToHom_map, assoc]
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
eq₁ :
  (H.hom ≫ H.inv).c.app (op ((Opens.map H.hom.base).obj U)) =
    (𝟙 X).c.app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom ⋯)
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
tactic: have eq₁ := congr_app H.hom_inv_id (op ((Opens.map H.hom.base).obj U))
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
eq₁ :
  (H.hom ≫ H.inv).c.app (op ((Opens.map H.hom.base).obj U)) =
    (𝟙 X).c.app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom ⋯)
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
eq₁ :
  (H.hom ≫ H.inv).c.app (op ((Opens.map H.hom.base).obj U)) =
    (𝟙 X).c.app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom ⋯)
eq₂ :
  Y.presheaf.map (eqToHom ⋯).op ≫ H.hom.c.app (op ((Opens.map ((forget C).map (𝟙 Y))).obj U)) =
    H.hom.c.app (op ((Opens.map ((forget C).map (H.inv ≫ H.hom))).obj U)) ≫
      ((Presheaf.pushforward C H.hom.base).obj X.presheaf).map (eqToHom ⋯).op
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
tactic: have eq₂ := H.hom.c.naturality (eqToHom (congr_obj (congr_arg Opens.map
      ((forget C).congr_map H.inv_hom_id.symm)) U)).op
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
eq₁ :
  (H.hom ≫ H.inv).c.app (op ((Opens.map H.hom.base).obj U)) =
    (𝟙 X).c.app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom ⋯)
eq₂ :
  Y.presheaf.map (eqToHom ⋯).op ≫ H.hom.c.app (op ((Opens.map ((forget C).map (𝟙 Y))).obj U)) =
    H.hom.c.app (op ((Opens.map ((forget C).map (H.inv ≫ H.hom))).obj U)) ≫
      ((Presheaf.pushforward C H.hom.base).obj X.presheaf).map (eqToHom ⋯).op
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
eq₁ :
  H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫
      H.hom.c.app (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))) =
    eqToHom ⋯
eq₂ :
  Y.presheaf.map (eqToHom ⋯).op ≫ H.hom.c.app (op ((Opens.map ((forget C).map (𝟙 Y))).obj U)) =
    H.hom.c.app (op ((Opens.map ((forget C).map (H.inv ≫ H.hom))).obj U)) ≫
      ((Presheaf.pushforward C H.hom.base).obj X.presheaf).map (eqToHom ⋯).op
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
tactic: rw [id_c, NatTrans.id_app, id_comp, eqToHom_map, comp_c_app] at eq₁
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
eq₁ :
  H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫
      H.hom.c.app (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))) =
    eqToHom ⋯
eq₂ :
  Y.presheaf.map (eqToHom ⋯).op ≫ H.hom.c.app (op ((Opens.map ((forget C).map (𝟙 Y))).obj U)) =
    H.hom.c.app (op ((Opens.map ((forget C).map (H.inv ≫ H.hom))).obj U)) ≫
      ((Presheaf.pushforward C H.hom.base).obj X.presheaf).map (eqToHom ⋯).op
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
eq₁ :
  H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫
      H.hom.c.app (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))) =
    eqToHom ⋯
eq₂ :
  eqToHom ⋯ ≫ H.hom.c.app (op ((Opens.map ((forget C).map (𝟙 Y))).obj U)) =
    H.hom.c.app (op ((Opens.map ((forget C).map (H.inv ≫ H.hom))).obj U)) ≫
      ((Presheaf.pushforward C H.hom.base).obj X.presheaf).map (eqToHom ⋯)
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
tactic: rw [eqToHom_op, eqToHom_map] at eq₂
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
eq₁ :
  H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫
      H.hom.c.app (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))) =
    eqToHom ⋯
eq₂ :
  eqToHom ⋯ ≫ H.hom.c.app (op ((Opens.map ((forget C).map (𝟙 Y))).obj U)) =
    H.hom.c.app (op ((Opens.map ((forget C).map (H.inv ≫ H.hom))).obj U)) ≫
      ((Presheaf.pushforward C H.hom.base).obj X.presheaf).map (eqToHom ⋯)
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
state_after: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
eq₁ :
  H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫
      H.hom.c.app (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))) =
    eqToHom ⋯
eq₂ :
  eqToHom ⋯ ≫ H.hom.c.app (op ((Opens.map ((forget C).map (𝟙 Y))).obj U)) =
    H.hom.c.app (op ((Opens.map ((forget C).map (H.inv ≫ H.hom))).obj U)) ≫
      ((Presheaf.pushforward C H.hom.base).obj X.presheaf).map (eqToHom ⋯)
⊢ eqToHom ⋯ ≫ ((Presheaf.pushforward C H.hom.base).obj X.presheaf).map (eqToHom ⋯) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
tactic: erw [eq₂, reassoc_of% eq₁]
state_before: case w
C : Type u_1
inst✝ : Category.{?u.38446, u_1} C
X Y : PresheafedSpace C
H : X ≅ Y
U : Opens ↑↑Y
eq₁ :
  H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫
      H.hom.c.app (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))) =
    eqToHom ⋯
eq₂ :
  eqToHom ⋯ ≫ H.hom.c.app (op ((Opens.map ((forget C).map (𝟙 Y))).obj U)) =
    H.hom.c.app (op ((Opens.map ((forget C).map (H.inv ≫ H.hom))).obj U)) ≫
      ((Presheaf.pushforward C H.hom.base).obj X.presheaf).map (eqToHom ⋯)
⊢ eqToHom ⋯ ≫ ((Presheaf.pushforward C H.hom.base).obj X.presheaf).map (eqToHom ⋯) =
    𝟙 (((Presheaf.pushforward C H.hom.base).obj X.presheaf).obj (op U))
state_after: no goals
tactic: simp