Module: Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks

Goal state before:
pp: case refine_2.refine_3
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↥(pullback f g) → ↥X × ↥Y := fun x ↦ ((pullback.fst f g) x, (pullback.snd f g) x)
H :
  ∀ (R A B : CommRingCat) (f' : Spec A ⟶ Spec R) (g' : Spec B ⟶ Spec R) (iX : Spec A ⟶ X) (iY : Spec B ⟶ Y)
    (iS : Spec R ⟶ S) (e₁ : f' ≫ iS = iX ≫ f) (e₂ : g' ≫ iS = iY ≫ g),
    IsOpenImmersion iX →
      IsOpenImmersion iY → IsOpenImmersion iS → IsEmbedding (L ∘ ⇑(pullback.map f' g' f g iX iY iS e₁ e₂))
𝒰 : S.OpenCover := S.affineOpenCover.openCover
𝒱 : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) → ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i).OpenCover :=
  fun i ↦ ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i).affineOpenCover.openCover
𝒲 : (i : (Precoverage.ZeroHypercover.pullback₁ g 𝒰).I₀) → ((Precoverage.ZeroHypercover.pullback₁ g 𝒰).X i).OpenCover :=
  fun i ↦ ((Precoverage.ZeroHypercover.pullback₁ g 𝒰).X i).affineOpenCover.openCover
U : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) × (𝒱 i).I₀ × (𝒲 i).I₀ → TopologicalSpace.Opens (↥X × ↥Y) :=
  fun ijk ↦
    {
      carrier :=
        {P |
          P.1 ∈ Scheme.Hom.opensRange ((𝒱 ijk.fst).f ijk.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f ijk.fst) ∧
            P.2 ∈
              Scheme.Hom.opensRange ((𝒲 ijk.fst).f ijk.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f ijk.fst)},
      is_open' := ⋯ }
this : Set.range L ⊆ ↑(iSup U)
𝓤 : (Scheme.precoverage IsOpenImmersion).ZeroHypercover (pullback f g) :=
  Precoverage.ZeroHypercover.bind (Scheme.Pullback.openCoverOfBase 𝒰 f g) fun i ↦
    Scheme.Pullback.openCoverOfLeftRight (𝒱 i) (𝒲 i) (pullback.snd f (𝒰.f i)) (pullback.snd g (𝒰.f i))
i : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) × (𝒱 i).I₀ × (𝒲 i).I₀
x : ↥(pullback f g)
x₁ : ↥((𝒱 i.fst).X i.snd.1)
hx₁ :
  ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst) x₁ =
    ((fun x ↦ ((pullback.fst f g) x, (pullback.snd f g) x)) x).1
x₂ : ↥((𝒲 i.fst).X i.snd.2)
hx₂ :
  ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst) x₂ =
    ((fun x ↦ ((pullback.fst f g) x, (pullback.snd f g) x)) x).2
x₁' : ↥(pullback (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
hx₁' :
  (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst)) x₁' = x
x₂' : ↥(pullback (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst))
hx₂' :
  (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)) x₂' = x
z :
  ↥(pullback
      (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
      (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)))
hz :
  (pullback.fst
        (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
        (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)))
      z =
    x₁'
⊢ (pullbackFstFstIso ((𝒱 i.fst).f i.snd.1 ≫ pullback.snd f (𝒰.f i.fst))
          ((𝒲 i.fst).f i.snd.2 ≫ pullback.snd g (𝒰.f i.fst)) f g
          ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst)
          ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst) (𝒰.f i.fst) ⋯ ⋯).hom ≫
      𝓤.f i =
    pullback.fst
        (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
        (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)) ≫
      pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst)

Tactic:
apply pullback.hom_ext

Goal state after:
case refine_2.refine_3.h₀
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↥(pullback f g) → ↥X × ↥Y := ⋯
H :
  ∀ (R A B : CommRingCat) (f' : Spec A ⟶ Spec R) (g' : Spec B ⟶ Spec R) (iX : Spec A ⟶ X) (iY : Spec B ⟶ Y)
    (iS : Spec R ⟶ S) (e₁ : f' ≫ iS = iX ≫ f) (e₂ : g' ≫ iS = iY ≫ g),
    IsOpenImmersion iX →
      IsOpenImmersion iY → IsOpenImmersion iS → IsEmbedding (L ∘ ⇑(pullback.map f' g' f g iX iY iS e₁ e₂))
𝒰 : S.OpenCover := ⋯
𝒱 : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) →
  ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i).OpenCover := ⋯
𝒲 : (i : (Precoverage.ZeroHypercover.pullback₁ g 𝒰).I₀) →
  ((Precoverage.ZeroHypercover.pullback₁ g 𝒰).X i).OpenCover := ⋯
U : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) × (𝒱 i).I₀ × (𝒲 i).I₀ → TopologicalSpace.Opens (↥X × ↥Y) := ⋯
this : Set.range L ⊆ ↑(iSup U)
𝓤 : (Scheme.precoverage IsOpenImmersion).ZeroHypercover (pullback f g) := ⋯
i : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) × (𝒱 i).I₀ × (𝒲 i).I₀
x : ↥(pullback f g)
x₁ : ↥((𝒱 i.fst).X i.snd.1)
hx₁ :
  ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst) x₁ =
    ((fun x ↦ ((pullback.fst f g) x, (pullback.snd f g) x)) x).1
x₂ : ↥((𝒲 i.fst).X i.snd.2)
hx₂ :
  ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst) x₂ =
    ((fun x ↦ ((pullback.fst f g) x, (pullback.snd f g) x)) x).2
x₁' : ↥(pullback (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
hx₁' :
  (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst)) x₁' = x
x₂' : ↥(pullback (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst))
hx₂' :
  (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)) x₂' = x
z :
  ↥(pullback
      (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
      (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)))
hz :
  (pullback.fst
        (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
        (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)))
      z =
    x₁'
⊢ ((pullbackFstFstIso ((𝒱 i.fst).f i.snd.1 ≫ pullback.snd f (𝒰.f i.fst))
            ((𝒲 i.fst).f i.snd.2 ≫ pullback.snd g (𝒰.f i.fst)) f g
            ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst)
            ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst) (𝒰.f i.fst) ⋯ ⋯).hom ≫
        𝓤.f i) ≫
      pullback.fst f g =
    (pullback.fst
          (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
          (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)) ≫
        pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst)) ≫
      pullback.fst f g

case h₁
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↥(pullback f g) → ↥X × ↥Y := ⋯
H :
  ∀ (R A B : CommRingCat) (f' : Spec A ⟶ Spec R) (g' : Spec B ⟶ Spec R) (iX : Spec A ⟶ X) (iY : Spec B ⟶ Y)
    (iS : Spec R ⟶ S) (e₁ : f' ≫ iS = iX ≫ f) (e₂ : g' ≫ iS = iY ≫ g),
    IsOpenImmersion iX →
      IsOpenImmersion iY → IsOpenImmersion iS → IsEmbedding (L ∘ ⇑(pullback.map f' g' f g iX iY iS e₁ e₂))
𝒰 : S.OpenCover := ⋯
𝒱 : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) →
  ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i).OpenCover := ⋯
𝒲 : (i : (Precoverage.ZeroHypercover.pullback₁ g 𝒰).I₀) →
  ((Precoverage.ZeroHypercover.pullback₁ g 𝒰).X i).OpenCover := ⋯
U : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) × (𝒱 i).I₀ × (𝒲 i).I₀ → TopologicalSpace.Opens (↥X × ↥Y) := ⋯
this : Set.range L ⊆ ↑(iSup U)
𝓤 : (Scheme.precoverage IsOpenImmersion).ZeroHypercover (pullback f g) := ⋯
i : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) × (𝒱 i).I₀ × (𝒲 i).I₀
x : ↥(pullback f g)
x₁ : ↥((𝒱 i.fst).X i.snd.1)
hx₁ :
  ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst) x₁ =
    ((fun x ↦ ((pullback.fst f g) x, (pullback.snd f g) x)) x).1
x₂ : ↥((𝒲 i.fst).X i.snd.2)
hx₂ :
  ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst) x₂ =
    ((fun x ↦ ((pullback.fst f g) x, (pullback.snd f g) x)) x).2
x₁' : ↥(pullback (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
hx₁' :
  (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst)) x₁' = x
x₂' : ↥(pullback (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst))
hx₂' :
  (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)) x₂' = x
z :
  ↥(pullback
      (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
      (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)))
hz :
  (pullback.fst
        (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
        (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)))
      z =
    x₁'
⊢ ((pullbackFstFstIso ((𝒱 i.fst).f i.snd.1 ≫ pullback.snd f (𝒰.f i.fst))
            ((𝒲 i.fst).f i.snd.2 ≫ pullback.snd g (𝒰.f i.fst)) f g
            ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst)
            ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst) (𝒰.f i.fst) ⋯ ⋯).hom ≫
        𝓤.f i) ≫
      pullback.snd f g =
    (pullback.fst
          (pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst))
          (pullback.fst (pullback.snd f g) ((𝒲 i.fst).f i.snd.2 ≫ (Precoverage.ZeroHypercover.pullback₁ g 𝒰).f i.fst)) ≫
        pullback.fst (pullback.fst f g) ((𝒱 i.fst).f i.snd.1 ≫ (Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i.fst)) ≫
      pullback.snd f g