mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Gluing

Goal state before:
pp: D : GlueData
h : MkCore
i : h.J
x✝ : ↑((Opens.toTopCat (h.U i)).obj (h.V i i))
⊢ id x✝ = (ConcreteCategory.hom (𝟙 ((Opens.toTopCat (h.U (i, i).1)).obj (h.V (i, i).1 (i, i).2)))) x✝

Tactic:
]

Goal state after:
D : GlueData
h : MkCore
i : h.J
x✝ : ↑((Opens.toTopCat (h.U i)).obj (h.V i i))
⊢ id x✝ = (ConcreteCategory.hom (𝟙 ((Opens.toTopCat (h.U (i, i).1)).obj (h.V (i, i).1 (i, i).2)))) x✝
Module: Mathlib.Topology.Gluing

Goal state before:
pp: D : GlueData
h : MkCore
i j k : h.J
⊢ ofHom
        {
          toFun := fun x ↦
            ⟨(⟨↑((ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩,
          continuous_toFun := ⋯ } ≫
      pullbackSnd (h.V j k).inclusion' (h.V j i).inclusion' =
    pullbackFst (h.V i j).inclusion' (h.V i k).inclusion' ≫ h.t i j

Tactic:
]

Goal state after:
D : GlueData
h : MkCore
i j k : h.J
⊢ ofHom
        {
          toFun := fun x ↦
            ⟨(⟨↑((ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩,
          continuous_toFun := ⋯ } ≫
      pullbackSnd (h.V j k).inclusion' (h.V j i).inclusion' =
    pullbackFst (h.V i j).inclusion' (h.V i k).inclusion' ≫ h.t i j
Module: Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits

Goal state before:
pp: X Y : RingedSpace
f g : X ⟶ Y
H : f = g
x : ↑↑X.toPresheafedSpace
h : IsLocalHom (CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap f.hom x))
⊢ IsLocalHom (CommRingCat.Hom.hom (eqToHom ⋯ ≫ PresheafedSpace.Hom.stalkMap f.hom x))

Tactic:
]

Goal state after:
X Y : RingedSpace
f g : X ⟶ Y
H : f = g
x : ↑↑X.toPresheafedSpace
h : IsLocalHom (CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap f.hom x))
⊢ IsLocalHom (CommRingCat.Hom.hom (eqToHom ⋯ ≫ PresheafedSpace.Hom.stalkMap f.hom x))
Module: Mathlib.Topology.Gluing

Goal state before:
pp: D : GlueData
h : MkCore
i j k : h.J
⊢ (((((((pullbackIsoProdSubtype (h.V i j).inclusion' (h.V i k).inclusion').hom ≫
                  ofHom
                    {
                      toFun := fun x ↦
                        ⟨(⟨↑((ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩,
                      continuous_toFun := ⋯ }) ≫
                (pullbackIsoProdSubtype (h.V j k).inclusion' (h.V j i).inclusion').inv) ≫
              (pullbackIsoProdSubtype (h.V j k).inclusion' (h.V j i).inclusion').hom) ≫
            ofHom
              {
                toFun := fun x ↦
                  ⟨(⟨↑((ConcreteCategory.hom (h.t j k)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t j k)) (↑x).1), ⋯⟩,
                continuous_toFun := ⋯ }) ≫
          (pullbackIsoProdSubtype (h.V k i).inclusion' (h.V k j).inclusion').inv) ≫
        (pullbackIsoProdSubtype (h.V k i).inclusion' (h.V k j).inclusion').hom) ≫
      ofHom
        {
          toFun := fun x ↦
            ⟨(⟨↑((ConcreteCategory.hom (h.t k i)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t k i)) (↑x).1), ⋯⟩,
          continuous_toFun := ⋯ } =
    𝟙 (pullback (h.V i j).inclusion' (h.V i k).inclusion') ≫
      (pullbackIsoProdSubtype (h.V i j).inclusion' (h.V i k).inclusion').hom

Tactic:
]

Goal state after:
D : GlueData
h : MkCore
i j k : h.J
⊢ (((((((pullbackIsoProdSubtype (h.V i j).inclusion' (h.V i k).inclusion').hom ≫
                  ofHom
                    {
                      toFun := fun x ↦
                        ⟨(⟨↑((ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩,
                      continuous_toFun := ⋯ }) ≫
                (pullbackIsoProdSubtype (h.V j k).inclusion' (h.V j i).inclusion').inv) ≫
              (pullbackIsoProdSubtype (h.V j k).inclusion' (h.V j i).inclusion').hom) ≫
            ofHom
              {
                toFun := fun x ↦
                  ⟨(⟨↑((ConcreteCategory.hom (h.t j k)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t j k)) (↑x).1), ⋯⟩,
                continuous_toFun := ⋯ }) ≫
          (pullbackIsoProdSubtype (h.V k i).inclusion' (h.V k j).inclusion').inv) ≫
        (pullbackIsoProdSubtype (h.V k i).inclusion' (h.V k j).inclusion').hom) ≫
      ofHom
        {
          toFun := fun x ↦
            ⟨(⟨↑((ConcreteCategory.hom (h.t k i)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t k i)) (↑x).1), ⋯⟩,
          continuous_toFun := ⋯ } =
    𝟙 (pullback (h.V i j).inclusion' (h.V i k).inclusion') ≫
      (pullbackIsoProdSubtype (h.V i j).inclusion' (h.V i k).inclusion').hom
Module: Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits

Goal state before:
pp: X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : Hom.toShHom f ≫ Hom.toShHom s.π = Hom.toShHom g ≫ Hom.toShHom s.π
y : ↑↑Y.toPresheafedSpace
h : ↑(s.pt.presheaf.stalk
      ((ConcreteCategory.hom (coequalizer.desc (Hom.toShHom s.π) e).hom.base)
        ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y))) →+*
  ↑((coequalizerCofork f g).pt.presheaf.stalk
      ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y)) :=
  CommRingCat.Hom.hom
    (PresheafedSpace.Hom.stalkMap (coequalizer.desc (Hom.toShHom s.π) e).hom
      ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y))
⊢ IsLocalHom (CommRingCat.Hom.hom (CommRingCat.ofHom h ≫ (coequalizerCofork f g).π.stalkMap y))

Tactic:
]

Goal state after:
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : Hom.toShHom f ≫ Hom.toShHom s.π = Hom.toShHom g ≫ Hom.toShHom s.π
y : ↑↑Y.toPresheafedSpace
h : ↑(s.pt.presheaf.stalk
      ((ConcreteCategory.hom (coequalizer.desc (Hom.toShHom s.π) e).hom.base)
        ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y))) →+*
  ↑((coequalizerCofork f g).pt.presheaf.stalk
      ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y)) :=
  CommRingCat.Hom.hom
    (PresheafedSpace.Hom.stalkMap (coequalizer.desc (Hom.toShHom s.π) e).hom
      ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y))
⊢ IsLocalHom (CommRingCat.Hom.hom (CommRingCat.ofHom h ≫ (coequalizerCofork f g).π.stalkMap y))
Module: Mathlib.Topology.Gluing

Goal state before:
pp: D : GlueData
h : MkCore
i j k : h.J
⊢ ofHom
        {
          toFun := fun x ↦
            ⟨(⟨↑((ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩,
          continuous_toFun := ⋯ } ≫
      ofHom
          {
            toFun := fun x ↦
              ⟨(⟨↑((ConcreteCategory.hom (h.t j k)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t j k)) (↑x).1), ⋯⟩,
            continuous_toFun := ⋯ } ≫
        ofHom
          {
            toFun := fun x ↦
              ⟨(⟨↑((ConcreteCategory.hom (h.t k i)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t k i)) (↑x).1), ⋯⟩,
            continuous_toFun := ⋯ } =
    𝟙 (of { p // (ConcreteCategory.hom (h.V i j).inclusion') p.1 = (ConcreteCategory.hom (h.V i k).inclusion') p.2 })

Tactic:
]

Goal state after:
D : GlueData
h : MkCore
i j k : h.J
⊢ ofHom
        {
          toFun := fun x ↦
            ⟨(⟨↑((ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩,
          continuous_toFun := ⋯ } ≫
      ofHom
          {
            toFun := fun x ↦
              ⟨(⟨↑((ConcreteCategory.hom (h.t j k)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t j k)) (↑x).1), ⋯⟩,
            continuous_toFun := ⋯ } ≫
        ofHom
          {
            toFun := fun x ↦
              ⟨(⟨↑((ConcreteCategory.hom (h.t k i)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t k i)) (↑x).1), ⋯⟩,
            continuous_toFun := ⋯ } =
    𝟙 (of { p // (ConcreteCategory.hom (h.V i j).inclusion') p.1 = (ConcreteCategory.hom (h.V i k).inclusion') p.2 })
Module: Mathlib.Topology.Gluing

Goal state before:
pp: D : GlueData
h : MkCore
i j k : h.J
x : ↑(h.U i)
hx : x ∈ h.V i j
hx' : x ∈ h.V i k
⊢ ↑((ConcreteCategory.hom (h.t k i))
        ⟨↑((ConcreteCategory.hom (h.t j k)) ⟨↑((ConcreteCategory.hom (h.t i j)) ⟨x, hx⟩), ⋯⟩), ⋯⟩) =
    x

Tactic:
]

Goal state after:
D : GlueData
h : MkCore
i j k : h.J
x : ↑(h.U i)
hx : x ∈ h.V i j
hx' : x ∈ h.V i k
⊢ ↑((ConcreteCategory.hom (h.t k i))
        ⟨↑((ConcreteCategory.hom (h.t j k)) ⟨↑((ConcreteCategory.hom (h.t i j)) ⟨x, hx⟩), ⋯⟩), ⋯⟩) =
    x
Module: Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits

Goal state before:
pp: X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : Hom.toShHom f ≫ Hom.toShHom s.π = Hom.toShHom g ≫ Hom.toShHom s.π
y : ↑↑Y.toPresheafedSpace
h : ↑(s.pt.presheaf.stalk
      ((ConcreteCategory.hom (coequalizer.desc (Hom.toShHom s.π) e).hom.base)
        ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y))) →+*
  ↑((coequalizerCofork f g).pt.presheaf.stalk
      ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y)) :=
  CommRingCat.Hom.hom
    (PresheafedSpace.Hom.stalkMap (coequalizer.desc (Hom.toShHom s.π) e).hom
      ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y))
⊢ IsLocalHom
    (CommRingCat.Hom.hom
      (PresheafedSpace.Hom.stalkMap
        ((coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom ≫ (coequalizer.desc (Hom.toShHom s.π) e).hom) y))

Tactic:
]

Goal state after:
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : Hom.toShHom f ≫ Hom.toShHom s.π = Hom.toShHom g ≫ Hom.toShHom s.π
y : ↑↑Y.toPresheafedSpace
h : ↑(s.pt.presheaf.stalk
      ((ConcreteCategory.hom (coequalizer.desc (Hom.toShHom s.π) e).hom.base)
        ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y))) →+*
  ↑((coequalizerCofork f g).pt.presheaf.stalk
      ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y)) :=
  CommRingCat.Hom.hom
    (PresheafedSpace.Hom.stalkMap (coequalizer.desc (Hom.toShHom s.π) e).hom
      ((ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) y))
⊢ IsLocalHom
    (CommRingCat.Hom.hom
      (PresheafedSpace.Hom.stalkMap
        ((coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom ≫ (coequalizer.desc (Hom.toShHom s.π) e).hom) y))
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Over

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X R : C
f : R ⟶ X
⊢ 𝟙 ((pullback f).obj (𝟙_ (Over X))).left = 𝟙 ((pullback f).obj (𝟙_ (Over X))).left

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Over

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X R✝ : C
f : R✝ ⟶ X
R S : Over X
⊢ Hom.left (𝟙 ((pullback f).obj (R ⊗ S))) ≫ pullback.fst (R ⊗ S).hom f ≫ pullback.fst R.hom S.hom =
    Hom.left (prodComparisonIso (pullback f) R S).hom ≫
      pullback.fst ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.fst R.hom f

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X R✝ : C
f : R✝ ⟶ X
R S : Over X
⊢ Hom.left (𝟙 ((pullback f).obj (R ⊗ S))) ≫ pullback.fst (R ⊗ S).hom f ≫ pullback.fst R.hom S.hom =
    Hom.left (prodComparisonIso (pullback f) R S).hom ≫
      pullback.fst ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.fst R.hom f
Module: Mathlib.Topology.Gluing

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
i : (ofOpenSubsets U).J
x : ↑(of α)
hx : x ∈ U i
j : (ofOpenSubsets U).J
y : ↑(of α)
hy : y ∈ U j
e : (ConcreteCategory.hom (U i).inclusion') ⟨x, hx⟩ = (ConcreteCategory.hom (U j).inclusion') ⟨y, hy⟩
⊢ (ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⟨x, hx⟩ = (ConcreteCategory.hom ((ofOpenSubsets U).ι j)) ⟨y, hy⟩

Tactic:
]

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
i : (ofOpenSubsets U).J
x : ↑(of α)
hx : x ∈ U i
j : (ofOpenSubsets U).J
y : ↑(of α)
hy : y ∈ U j
e : (ConcreteCategory.hom (U i).inclusion') ⟨x, hx⟩ = (ConcreteCategory.hom (U j).inclusion') ⟨y, hy⟩
⊢ (ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⟨x, hx⟩ = (ConcreteCategory.hom ((ofOpenSubsets U).ι j)) ⟨y, hy⟩
Module: Mathlib.Topology.Gluing

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
i : (ofOpenSubsets U).J
x : ↑(of α)
hx : x ∈ U i
j : (ofOpenSubsets U).J
hy : (ConcreteCategory.hom (U i).inclusion') ⟨x, hx⟩ ∈ U j
⊢ (ofOpenSubsets U).Rel ⟨i, ⟨x, hx⟩⟩ ⟨j, ⟨(ConcreteCategory.hom (U i).inclusion') ⟨x, hx⟩, hy⟩⟩

Tactic:
]

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
i : (ofOpenSubsets U).J
x : ↑(of α)
hx : x ∈ U i
j : (ofOpenSubsets U).J
hy : (ConcreteCategory.hom (U i).inclusion') ⟨x, hx⟩ ∈ U j
⊢ (ofOpenSubsets U).Rel ⟨i, ⟨x, hx⟩⟩ ⟨j, ⟨(ConcreteCategory.hom (U i).inclusion') ⟨x, hx⟩, hy⟩⟩
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Over

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X R✝ : C
f : R✝ ⟶ X
R S : Over X
⊢ Hom.left (𝟙 ((pullback f).obj (R ⊗ S))) ≫ pullback.fst (R ⊗ S).hom f ≫ pullback.snd R.hom S.hom =
    Hom.left (prodComparisonIso (pullback f) R S).hom ≫
      pullback.snd ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.fst S.hom f

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X R✝ : C
f : R✝ ⟶ X
R S : Over X
⊢ Hom.left (𝟙 ((pullback f).obj (R ⊗ S))) ≫ pullback.fst (R ⊗ S).hom f ≫ pullback.snd R.hom S.hom =
    Hom.left (prodComparisonIso (pullback f) R S).hom ≫
      pullback.snd ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.fst S.hom f
Module: Mathlib.Topology.Gluing

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
⊢ IsOpen[inst✝] (⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s)

Tactic:
]

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
⊢ IsOpen[inst✝] (⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s)
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Over

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X R✝ : C
f : R✝ ⟶ X
R S : Over X
⊢ Hom.left (𝟙 ((pullback f).obj (R ⊗ S))) ≫ pullback.snd (R ⊗ S).hom f =
    Hom.left (prodComparisonIso (pullback f) R S).hom ≫
      pullback.snd ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.snd S.hom f

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X R✝ : C
f : R✝ ⟶ X
R S : Over X
⊢ Hom.left (𝟙 ((pullback f).obj (R ⊗ S))) ≫ pullback.snd (R ⊗ S).hom f =
    Hom.left (prodComparisonIso (pullback f) R S).hom ≫
      pullback.snd ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.snd S.hom f
Module: Mathlib.Topology.Gluing

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
⊢ ∀ x ∈ ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s,
    ∃ t ⊆ ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s, IsOpen[inst✝] t ∧ x ∈ t

Tactic:
]

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
⊢ ∀ x ∈ ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s,
    ∃ t ⊆ ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s, IsOpen[inst✝] t ∧ x ∈ t
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Over

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X Y : C
f : X ⟶ Y
A B : Over Y
⊢ pullback.fst (pullback.fst A.hom B.hom ≫ A.hom) f ≫ pullback.fst A.hom B.hom =
    Hom.left (prodComparisonIso (pullback f) A B).hom ≫
      pullback.fst (pullback.snd A.hom f) (pullback.snd B.hom f) ≫ pullback.fst A.hom f

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X Y : C
f : X ⟶ Y
A B : Over Y
⊢ pullback.fst (pullback.fst A.hom B.hom ≫ A.hom) f ≫ pullback.fst A.hom B.hom =
    Hom.left (prodComparisonIso (pullback f) A B).hom ≫
      pullback.fst (pullback.snd A.hom f) (pullback.snd B.hom f) ≫ pullback.fst A.hom f
Module: Mathlib.Topology.Gluing

Goal state before:
pp: case right.left
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
i : (ofOpenSubsets U).J
x : ↑(of α)
hx' : x ∈ U i
hx : (ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⟨x, hx'⟩ ∈ s
⊢ IsOpen[inst✝]
    (⇑(ConcreteCategory.hom (U i).inclusion') ''
      ⇑(ConcreteCategory.hom (U i).inclusion') ⁻¹' ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s)

Tactic:
]

Goal state after:
case right.left
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
i : (ofOpenSubsets U).J
x : ↑(of α)
hx' : x ∈ U i
hx : (ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⟨x, hx'⟩ ∈ s
⊢ IsOpen[inst✝]
    (⇑(ConcreteCategory.hom (U i).inclusion') ''
      ⇑(ConcreteCategory.hom (U i).inclusion') ⁻¹' ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s)
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Over

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X A B Y : C
f : X ⟶ Y
gA : A ⟶ Y
gB : B ⟶ Y
⊢ pullback.fst (pullback.fst gA gB ≫ gA) f ≫ pullback.snd gA gB =
    Hom.left (prodComparisonIso (pullback f) (mk gA) (mk gB)).hom ≫
      pullback.snd ((pullback f).obj (mk gA)).hom ((pullback f).obj (mk gB)).hom ≫ pullback.fst (mk gB).hom f

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X A B Y : C
f : X ⟶ Y
gA : A ⟶ Y
gB : B ⟶ Y
⊢ pullback.fst (pullback.fst gA gB ≫ gA) f ≫ pullback.snd gA gB =
    Hom.left (prodComparisonIso (pullback f) (mk gA) (mk gB)).hom ≫
      pullback.snd ((pullback f).obj (mk gA)).hom ((pullback f).obj (mk gB)).hom ≫ pullback.fst (mk gB).hom f
Module: Mathlib.Topology.Gluing

Goal state before:
pp: case e'_3
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
i : (ofOpenSubsets U).J
x : ↑(of α)
hx' : x ∈ U i
hx : (ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⟨x, hx'⟩ ∈ s
e_1✝ : ↑((Opens.toTopCat (of α)).obj (U i)) = ↑((ofOpenSubsets U).U i)
⊢ ⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹'
      ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) ⁻¹' ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s =
    ⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s

Tactic:
]

Goal state after:
case e'_3
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
i : (ofOpenSubsets U).J
x : ↑(of α)
hx' : x ∈ U i
hx : (ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⟨x, hx'⟩ ∈ s
e_1✝ : ↑((Opens.toTopCat (of α)).obj (U i)) = ↑((ofOpenSubsets U).U i)
⊢ ⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹'
      ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) ⁻¹' ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s =
    ⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s
Module: Mathlib.Topology.Gluing

Goal state before:
pp: case right.right
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
i : (ofOpenSubsets U).J
x : ↑(of α)
hx' : x ∈ U i
hx : (ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⟨x, hx'⟩ ∈ s
⊢ (ConcreteCategory.hom (U i).inclusion') ⟨x, hx'⟩ ∈ Set.range ⇑(ConcreteCategory.hom (U i).inclusion')

Tactic:
]

Goal state after:
case right.right
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
s : Set ↑(ofOpenSubsets U).glued
hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)
i : (ofOpenSubsets U).J
x : ↑(of α)
hx' : x ∈ U i
hx : (ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⟨x, hx'⟩ ∈ s
⊢ (ConcreteCategory.hom (U i).inclusion') ⟨x, hx'⟩ ∈ Set.range ⇑(ConcreteCategory.hom (U i).inclusion')
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Over

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X A B Y : C
f : X ⟶ Y
gA : A ⟶ Y
gB : B ⟶ Y
⊢ pullback.snd (pullback.fst gA gB ≫ gA) f =
    Hom.left (prodComparisonIso (pullback f) (mk gA) (mk gB)).hom ≫
      pullback.snd ((pullback f).obj (mk gA)).hom ((pullback f).obj (mk gB)).hom ≫ pullback.snd (mk gB).hom f

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
X A B Y : C
f : X ⟶ Y
gA : A ⟶ Y
gB : B ⟶ Y
⊢ pullback.snd (pullback.fst gA gB ≫ gA) f =
    Hom.left (prodComparisonIso (pullback f) (mk gA) (mk gB)).hom ≫
      pullback.snd ((pullback f).obj (mk gA)).hom ((pullback f).obj (mk gB)).hom ≫ pullback.snd (mk gB).hom f
Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
⊢ IsOpenEmbedding
    (⇑(ConcreteCategory.hom (D.gluedIso (forget C)).inv) ∘ ⇑(ConcreteCategory.hom ((D.mapGlueData (forget C)).ι i)))

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
⊢ IsOpenEmbedding
    (⇑(ConcreteCategory.hom (D.gluedIso (forget C)).inv) ∘ ⇑(ConcreteCategory.hom ((D.mapGlueData (forget C)).ι i)))
Module: Mathlib.AlgebraicGeometry.Gluing

Goal state before:
pp: D : GlueData
i : D.J
⊢ LocallyRingedSpace.IsOpenImmersion (D.toLocallyRingedSpaceGlueData.ι i ≫ D.isoLocallyRingedSpace.inv)

Tactic:
]

Goal state after:
D : GlueData
i : D.J
⊢ LocallyRingedSpace.IsOpenImmersion (D.toLocallyRingedSpaceGlueData.ι i ≫ D.isoLocallyRingedSpace.inv)
Module: Mathlib.Topology.Gluing

Goal state before:
pp: case mp
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
i : (ofOpenSubsets U).J
x : ↑(of α)
hx' : x ∈ U i
⊢ (ConcreteCategory.hom (U i).inclusion') ⟨x, hx'⟩ ∈ ⋃ i, ↑(U i)

Tactic:
]

Goal state after:
case mp
α : Type u
inst✝ : TopologicalSpace α
J : Type u
U : J → Opens α
i : (ofOpenSubsets U).J
x : ↑(of α)
hx' : x ∈ U i
⊢ (ConcreteCategory.hom (U i).inclusion') ⟨x, hx'⟩ ∈ ⋃ i, ↑(U i)