-- Mathlib/AlgebraicGeometry/ValuativeCriterion.lean
@[stacks 01KZ]
lemma IsSeparated.valuativeCriterion [IsSeparated f] : ValuativeCriterion.Uniqueness f

Proof state:
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this✝ : IsClosedImmersion g
hZ : IsAffine Z
l : Spec (CommRingCat.of S.K) ⟶ Z := pullback.lift S.i₁ (Spec.map (CommRingCat.ofHom (algebraMap S.R S.K))) ⋯
hg : l ≫ g = Spec.map (CommRingCat.ofHom (algebraMap S.R S.K))
e : Arrow.mk (CommRingCat.ofHom (algebraMap S.R S.K)) ≅
  Arrow.mk (Scheme.Hom.appTop (Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)))) :=
  arrowIsoΓSpecOfIsAffine (CommRingCat.ofHom (algebraMap S.R S.K))
P : MorphismProperty CommRingCat :=
  RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] f => Function.Injective ⇑f
this : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] f => Function.Injective ⇑f).RespectsIso
⊢ P (CommRingCat.ofHom (algebraMap S.R S.K))

Tactic:
exact FaithfulSMul.algebraMap_injective S.R S.K

Result:
no goals

Proof state:
case allEq.mk.mk.l.left
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this✝ : IsClosedImmersion g
hZ : IsAffine Z
l : Spec (CommRingCat.of S.K) ⟶ Z := pullback.lift S.i₁ (Spec.map (CommRingCat.ofHom (algebraMap S.R S.K))) ⋯
hg : l ≫ g = Spec.map (CommRingCat.ofHom (algebraMap S.R S.K))
this : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop g ≫ Scheme.Hom.appTop l))
⊢ Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop g))

Tactic:
exact Function.Injective.of_comp this

Result:
no goals

Proof state:
case h₀
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this : IsClosedImmersion g
hZ : IsAffine Z
⊢ (S.i₁ ≫ pullback.diagonal f) ≫ pullback.fst f f =
    (Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ pullback.lift l₁ l₂ h) ≫ pullback.fst f f

case h₁
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this : IsClosedImmersion g
hZ : IsAffine Z
⊢ (S.i₁ ≫ pullback.diagonal f) ≫ pullback.snd f f =
    (Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ pullback.lift l₁ l₂ h) ≫ pullback.snd f f

Tactic:
simpa using hl₁.symm

Result:
case h₁
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this : IsClosedImmersion g
hZ : IsAffine Z
⊢ (S.i₁ ≫ pullback.diagonal f) ≫ pullback.snd f f =
    (Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ pullback.lift l₁ l₂ h) ≫ pullback.snd f f

Proof state:
case allEq.mk.mk.l
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
⊢ l₁ = l₂

Tactic:
have : IsClosedImmersion g := MorphismProperty.pullback_snd _ _ inferInstance

Result:
case allEq.mk.mk.l
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this : IsClosedImmersion g
⊢ l₁ = l₂

Proof state:
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h✝ : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h✝)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h✝)
this : IsClosedImmersion g
hZ : IsAffine Z
h : Function.Bijective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop g))
⊢ IsAffine Z ∧ IsIso (Scheme.Hom.appTop g)

Tactic:
exact ⟨hZ, (ConcreteCategory.isIso_iff_bijective _).mpr h⟩

Result:
no goals

Proof state:
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this : IsClosedImmersion g
hZ : IsAffine Z
⊢ Spec (CommRingCat.of S.K) ⟶ Z

Tactic:
apply pullback.lift S.i₁ (Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)))

Result:
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this : IsClosedImmersion g
hZ : IsAffine Z
⊢ S.i₁ ≫ pullback.diagonal f = Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ pullback.lift l₁ l₂ h