Theorem: AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_snd
File: Mathlib/Geometry/RingedSpace/OpenImmersion.lean

tactic: refine PresheafedSpace.Hom.ext _ _ ?_ <| NatTrans.ext _ _ <| funext fun x => ?_
state_before: C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).snd = s.snd
state_after: case refine_1
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).snd).base = s.snd.base

case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ ((pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).snd).c ≫ whiskerRight (eqToHom ⋯) s.pt.presheaf).app x =
    s.snd.c.app x

tactic: change pullback.lift _ _ _ ≫ pullback.snd = _
state_before: case refine_1
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).snd).base = s.snd.base
state_after: case refine_1
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ pullback.lift s.fst.base s.snd.base ⋯ ≫ pullback.snd = s.snd.base

tactic: simp
state_before: case refine_1
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ pullback.lift s.fst.base s.snd.base ⋯ ≫ pullback.snd = s.snd.base
state_after: no goals

tactic: change (_ ≫ _ ≫ _) ≫ _ = _
state_before: case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ ((pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).snd).c ≫ whiskerRight (eqToHom ⋯) s.pt.presheaf).app x =
    s.snd.c.app x
state_after: case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ ((pullbackConeOfLeft f g).snd.c.app x ≫
        s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).snd.base).op.obj x)) ≫
          s.pt.presheaf.map (eqToHom ⋯)) ≫
      (whiskerRight (eqToHom ⋯) s.pt.presheaf).app x =
    s.snd.c.app x

tactic: simp_rw [Category.assoc]
state_before: case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ ((pullbackConeOfLeft f g).snd.c.app x ≫
        s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).snd.base).op.obj x)) ≫
          s.pt.presheaf.map (eqToHom ⋯)) ≫
      (whiskerRight (eqToHom ⋯) s.pt.presheaf).app x =
    s.snd.c.app x
state_after: case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ (pullbackConeOfLeft f g).snd.c.app x ≫
      s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).snd.base).op.obj x)) ≫
        s.pt.presheaf.map (eqToHom ⋯) ≫ (whiskerRight (eqToHom ⋯) s.pt.presheaf).app x =
    s.snd.c.app x

tactic: erw [s.snd.c.naturality_assoc]
state_before: case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ (pullbackConeOfLeft f g).snd.c.app x ≫
      s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).snd.base).op.obj x)) ≫
        s.pt.presheaf.map (eqToHom ⋯) ≫ (whiskerRight (eqToHom ⋯) s.pt.presheaf).app x =
    s.snd.c.app x
state_after: case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ s.snd.c.app x ≫
      (s.snd.base _* s.pt.presheaf).map (⋯.adjunction.counit.app x.unop).op ≫
        s.pt.presheaf.map (eqToHom ⋯) ≫ (whiskerRight (eqToHom ⋯) s.pt.presheaf).app x =
    s.snd.c.app x

tactic: erw [← s.pt.presheaf.map_comp, ← s.pt.presheaf.map_comp]
state_before: case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ s.snd.c.app x ≫
      (s.snd.base _* s.pt.presheaf).map (⋯.adjunction.counit.app x.unop).op ≫
        s.pt.presheaf.map (eqToHom ⋯) ≫ (whiskerRight (eqToHom ⋯) s.pt.presheaf).app x =
    s.snd.c.app x
state_after: case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ s.snd.c.app x ≫
      s.pt.presheaf.map
        ((Opens.map s.snd.base).op.map (⋯.adjunction.counit.app x.unop).op ≫ eqToHom ⋯ ≫ (eqToHom ⋯).app x) =
    s.snd.c.app x

tactic: trans s.snd.c.app x ≫ s.pt.presheaf.map (𝟙 _)
state_before: case refine_2
C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ s.snd.c.app x ≫
      s.pt.presheaf.map
        ((Opens.map s.snd.base).op.map (⋯.adjunction.counit.app x.unop).op ≫ eqToHom ⋯ ≫ (eqToHom ⋯).app x) =
    s.snd.c.app x
state_after: C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ s.snd.c.app x ≫
      s.pt.presheaf.map
        ((Opens.map s.snd.base).op.map (⋯.adjunction.counit.app x.unop).op ≫ eqToHom ⋯ ≫ (eqToHom ⋯).app x) =
    s.snd.c.app x ≫ s.pt.presheaf.map (𝟙 ((Opens.map s.snd.base).op.obj x))

C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ s.snd.c.app x ≫ s.pt.presheaf.map (𝟙 ((Opens.map s.snd.base).op.obj x)) = s.snd.c.app x

tactic: congr 1
state_before: C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ s.snd.c.app x ≫
      s.pt.presheaf.map
        ((Opens.map s.snd.base).op.map (⋯.adjunction.counit.app x.unop).op ≫ eqToHom ⋯ ≫ (eqToHom ⋯).app x) =
    s.snd.c.app x ≫ s.pt.presheaf.map (𝟙 ((Opens.map s.snd.base).op.obj x))
state_after: no goals

tactic: rw [s.pt.presheaf.map_id]
state_before: C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ s.snd.c.app x ≫ s.pt.presheaf.map (𝟙 ((Opens.map s.snd.base).op.obj x)) = s.snd.c.app x
state_after: C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ s.snd.c.app x ≫ 𝟙 (s.pt.presheaf.obj ((Opens.map s.snd.base).op.obj x)) = s.snd.c.app x

tactic: erw [Category.comp_id]
state_before: C : Type u
inst✝ : Category.{v, u} C
X Y Z : PresheafedSpace C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens ↑↑Y)ᵒᵖ
⊢ s.snd.c.app x ≫ 𝟙 (s.pt.presheaf.obj ((Opens.map s.snd.base).op.obj x)) = s.snd.c.app x
state_after: no goals