Theorem: AlgebraicGeometry.PresheafedSpace.stalkMap.stalkSpecializes_stalkMap
File: Mathlib/Geometry/RingedSpace/Stalks.lean

Proof steps:
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
⊢ Y.presheaf.stalkSpecializes ⋯ ≫ Hom.stalkMap f x = Hom.stalkMap f y ≫ X.presheaf.stalkSpecializes h
state_after: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
⊢ colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)
        { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),
          ι :=
            {
              app := fun U =>
                colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf) (op { obj := (unop U).obj, property := ⋯ }),
              naturality := ⋯ } } ≫
      colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫
        colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
          colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =
    (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫
        colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫
          colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫
      colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)
        { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),
          ι :=
            {
              app := fun U =>
                colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),
              naturality := ⋯ } }
tactic: dsimp [stalkSpecializes, Hom.stalkMap, stalkFunctor, stalkPushforward]
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
⊢ colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)
        { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),
          ι :=
            {
              app := fun U =>
                colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf) (op { obj := (unop U).obj, property := ⋯ }),
              naturality := ⋯ } } ≫
      colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫
        colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
          colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =
    (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫
        colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫
          colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫
      colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)
        { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),
          ι :=
            {
              app := fun U =>
                colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),
              naturality := ⋯ } }
state_after: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : (OpenNhds (f.base y))ᵒᵖ
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj
          Y.presheaf)
        j ≫
      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)
          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)
                    (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } } ≫
        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =
    colimit.ι
        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj
          Y.presheaf)
        j ≫
      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫
        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)
          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } }
tactic: refine colimit.hom_ext fun j => ?_
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : (OpenNhds (f.base y))ᵒᵖ
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj
          Y.presheaf)
        j ≫
      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)
          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)
                    (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } } ≫
        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =
    colimit.ι
        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj
          Y.presheaf)
        j ≫
      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫
        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)
          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } }
state_after: case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (f.base y)
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj
          Y.presheaf)
        (op j) ≫
      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)
          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)
                    (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } } ≫
        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =
    colimit.ι
        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj
          Y.presheaf)
        (op j) ≫
      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫
        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)
          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } }
tactic: induction j with | h j => ?_
state_before: case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (f.base y)
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj
          Y.presheaf)
        (op j) ≫
      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)
          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)
                    (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } } ≫
        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =
    colimit.ι
        (((whiskeringLeft (OpenNhds (f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f.base y)).op).obj
          Y.presheaf)
        (op j) ≫
      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫
        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)
          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } }
state_after: case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (f.base y)
⊢ colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫
      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)
          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)
                    (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } } ≫
        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =
    colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫
      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫
        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)
          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } }
tactic: dsimp
state_before: case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (f.base y)
⊢ colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫
      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)
          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)
                    (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } } ≫
        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =
    colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫
      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫
        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)
          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),
            ι :=
              {
                app := fun U =>
                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),
                naturality := ⋯ } }
state_after: case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (f.base y)
⊢ f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫
      X.presheaf.map (𝟙 ((OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op.obj (op { obj := j.obj, property := ⋯ }))) ≫
        colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)
          ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) =
    f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫
      X.presheaf.map (𝟙 ((OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op.obj (op j))) ≫
        colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)
          (op { obj := (unop ((OpenNhds.map f.base y).op.obj (op j))).obj, property := ⋯ })
tactic: simp only [colimit.ι_desc_assoc, ι_colimMap_assoc, whiskerLeft_app,
    whiskerRight_app, NatTrans.id_app, map_id, colimit.ι_pre, id_comp, assoc,
    colimit.pre_desc, colimit.map_desc, colimit.ι_desc, Cocones.precompose_obj_ι,
    Cocone.whisker_ι, NatTrans.comp_app]
state_before: case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (f.base y)
⊢ f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫
      X.presheaf.map (𝟙 ((OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op.obj (op { obj := j.obj, property := ⋯ }))) ≫
        colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)
          ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) =
    f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫
      X.presheaf.map (𝟙 ((OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op.obj (op j))) ≫
        colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)
          (op { obj := (unop ((OpenNhds.map f.base y).op.obj (op j))).obj, property := ⋯ })
state_after: case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (f.base y)
⊢ f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)
        ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) =
    f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)
        ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ }))
tactic: erw [X.presheaf.map_id, id_comp]
state_before: case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (f.base y)
⊢ f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)
        ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) =
    f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)
        ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ }))
state_after: no goals
tactic: rfl