-- Mathlib/AlgebraicGeometry/Morphisms/SurjectiveOnStalks.lean
/-- If `Y ⟶ S` is surjective on stalks, then for every `X ⟶ S`, `X ×ₛ Y` is a subset of
`X × Y` (cartesian product as topological spaces) with the induced topology. -/
lemma isEmbedding_pullback {X Y S : Scheme.{u}} (f : X ⟶ S) (g : Y ⟶ S) [SurjectiveOnStalks g] :
    IsEmbedding (fun x ↦ ((pullback.fst f g).base x, (pullback.snd f g).base x))

Proof state:
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
  fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategory.hom (pullback.snd f g).base) 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 ∘ ⇑(ConcreteCategory.hom (pullback.map f' g' f g iX iY iS e₁ e₂).base))
𝒰 : S.OpenCover := S.affineOpenCover.openCover
𝒱 : (i : (Scheme.Cover.pullbackCover 𝒰 f).J) → ((Scheme.Cover.pullbackCover 𝒰 f).obj i).OpenCover :=
  fun i => ((Scheme.Cover.pullbackCover 𝒰 f).obj i).affineOpenCover.openCover
𝒲 : (i : (Scheme.Cover.pullbackCover 𝒰 g).J) → ((Scheme.Cover.pullbackCover 𝒰 g).obj i).OpenCover :=
  fun i => ((Scheme.Cover.pullbackCover 𝒰 g).obj i).affineOpenCover.openCover
U : (i : (Scheme.Cover.pullbackCover 𝒰 f).J) × (𝒱 i).J × (𝒲 i).J →
  TopologicalSpace.Opens (↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace) :=
  fun ijk =>
    {
      carrier :=
        {P |
          P.1 ∈ Scheme.Hom.opensRange ((𝒱 ijk.fst).map ijk.snd.1 ≫ (Scheme.Cover.pullbackCover 𝒰 f).map ijk.fst) ∧
            P.2 ∈ Scheme.Hom.opensRange ((𝒲 ijk.fst).map ijk.snd.2 ≫ (Scheme.Cover.pullbackCover 𝒰 g).map ijk.fst)},
      is_open' := ⋯ }
this : Set.range L ⊆ ↑(iSup U)
𝓤 : Scheme.Cover (@IsOpenImmersion) (pullback f g) :=
  Scheme.Cover.bind (Scheme.Pullback.openCoverOfBase 𝒰 f g) fun i =>
    Scheme.Pullback.openCoverOfLeftRight (𝒱 i) (𝒲 i) (pullback.snd f (𝒰.map i)) (pullback.snd g (𝒰.map i))
i : (i : (Scheme.Cover.pullbackCover 𝒰 f).J) × (𝒱 i).J × (𝒲 i).J
⊢ ((𝒲 i.fst).map i.snd.2 ≫ Scheme.Cover.pullbackHom 𝒰 g i.fst) ≫ 𝒰.map i.fst =
    ((𝒲 i.fst).map i.snd.2 ≫ (Scheme.Cover.pullbackCover 𝒰 g).map i.fst) ≫ g

Tactic:
simp [pullback.condition]

Result:
no goals

Proof state:
case intro.intro
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
  fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategory.hom (pullback.snd f g).base) x)
R A B : CommRingCat
iX : Spec A ⟶ X
iY : Spec B ⟶ Y
iS : Spec R ⟶ S
a✝² : IsOpenImmersion iX
a✝¹ : IsOpenImmersion iY
a✝ : IsOpenImmersion iS
φ : R ⟶ A
e₁ : Spec.map φ ≫ iS = iX ≫ f
ψ : R ⟶ B
e₂ : Spec.map ψ ≫ iS = iY ≫ g
H : SurjectiveOnStalks (Spec.map ψ)
algInst✝¹ : Algebra ↑R ↑A := (CommRingCat.Hom.hom φ).toAlgebra
algInst✝ : Algebra ↑R ↑B := (CommRingCat.Hom.hom ψ).toAlgebra
⊢ IsEmbedding (L ∘ ⇑(ConcreteCategory.hom (pullback.map (Spec.map φ) (Spec.map ψ) f g iX iY iS e₁ e₂).base))

Tactic:
rw [HasRingHomProperty.Spec_iff (P := @SurjectiveOnStalks)] at H

Result:
case intro.intro
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
  fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategory.hom (pullback.snd f g).base) x)
R A B : CommRingCat
iX : Spec A ⟶ X
iY : Spec B ⟶ Y
iS : Spec R ⟶ S
a✝² : IsOpenImmersion iX
a✝¹ : IsOpenImmersion iY
a✝ : IsOpenImmersion iS
φ : R ⟶ A
e₁ : Spec.map φ ≫ iS = iX ≫ f
ψ : R ⟶ B
e₂ : Spec.map ψ ≫ iS = iY ≫ g
H : (CommRingCat.Hom.hom ψ).SurjectiveOnStalks
algInst✝¹ : Algebra ↑R ↑A := (CommRingCat.Hom.hom φ).toAlgebra
algInst✝ : Algebra ↑R ↑B := (CommRingCat.Hom.hom ψ).toAlgebra
⊢ IsEmbedding (L ∘ ⇑(ConcreteCategory.hom (pullback.map (Spec.map φ) (Spec.map ψ) f g iX iY iS e₁ e₂).base))

Proof state:
case intro.intro
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
  fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategory.hom (pullback.snd f g).base) x)
R A B : CommRingCat
iX : Spec A ⟶ X
iY : Spec B ⟶ Y
iS : Spec R ⟶ S
a✝² : IsOpenImmersion iX
a✝¹ : IsOpenImmersion iY
a✝ : IsOpenImmersion iS
φ : R ⟶ A
e₁ : Spec.map φ ≫ iS = iX ≫ f
ψ : R ⟶ B
e₂ : Spec.map ψ ≫ iS = iY ≫ g
H : SurjectiveOnStalks (Spec.map ψ)
⊢ IsEmbedding (L ∘ ⇑(ConcreteCategory.hom (pullback.map (Spec.map φ) (Spec.map ψ) f g iX iY iS e₁ e₂).base))

Tactic:
algebraize [φ.hom, ψ.hom]

Result:
case intro.intro
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
  fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategory.hom (pullback.snd f g).base) x)
R A B : CommRingCat
iX : Spec A ⟶ X
iY : Spec B ⟶ Y
iS : Spec R ⟶ S
a✝² : IsOpenImmersion iX
a✝¹ : IsOpenImmersion iY
a✝ : IsOpenImmersion iS
φ : R ⟶ A
e₁ : Spec.map φ ≫ iS = iX ≫ f
ψ : R ⟶ B
e₂ : Spec.map ψ ≫ iS = iY ≫ g
H : SurjectiveOnStalks (Spec.map ψ)
algInst✝¹ : Algebra ↑R ↑A := (CommRingCat.Hom.hom φ).toAlgebra
algInst✝ : Algebra ↑R ↑B := (CommRingCat.Hom.hom ψ).toAlgebra
⊢ IsEmbedding (L ∘ ⇑(ConcreteCategory.hom (pullback.map (Spec.map φ) (Spec.map ψ) f g iX iY iS e₁ e₂).base))