Theorem: AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk
File: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ awayToΓ 𝒜 f ≫ (Proj.restrict ⋯).presheaf.Γgerm x =
    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv ≫ (Proj.restrictStalkIso ⋯ x).inv
state_after: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ awayToSection 𝒜 f ≫
      (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫
        (Proj.restrict ⋯).presheaf.germ ⟨x, ⋯⟩ ≫ (Proj.restrictStalkIso ⋯ x).hom =
    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv
tactic: rw [awayToΓ, Category.assoc, ← Category.assoc _ (Iso.inv _),
    Iso.eq_comp_inv, Category.assoc, Category.assoc, Presheaf.Γgerm]
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ awayToSection 𝒜 f ≫
      (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫
        (Proj.restrict ⋯).presheaf.germ ⟨x, ⋯⟩ ≫ (Proj.restrictStalkIso ⋯ x).hom =
    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv
state_after: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ awayToSection 𝒜 f ≫ (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ Proj.presheaf.germ ⟨(pbo f).inclusion x, ⋯⟩ =
    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv
tactic: rw [LocallyRingedSpace.restrictStalkIso_hom_eq_germ]
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ awayToSection 𝒜 f ≫ (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ Proj.presheaf.germ ⟨(pbo f).inclusion x, ⋯⟩ =
    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv
state_after: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ awayToSection 𝒜 f ≫
      (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ Presheaf.germ (structureSheaf 𝒜).val ⟨(pbo f).inclusion x, ⋯⟩ =
    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv
tactic: simp only [Proj.toLocallyRingedSpace, Proj.toSheafedSpace]
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ awayToSection 𝒜 f ≫
      (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ Presheaf.germ (structureSheaf 𝒜).val ⟨(pbo f).inclusion x, ⋯⟩ =
    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv
state_after: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ HomogeneousLocalization.mapId 𝒜 ⋯ ≫
      (Proj.stalkIso' 𝒜 ↑((fun x => ⟨↑x, ⋯⟩) ⟨(pbo f).inclusion x, ⋯⟩)).toCommRingCatIso.inv =
    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv
tactic: rw [Presheaf.germ_res, awayToSection_germ]
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑↑(Proj.restrict ⋯).toPresheafedSpace
⊢ HomogeneousLocalization.mapId 𝒜 ⋯ ≫
      (Proj.stalkIso' 𝒜 ↑((fun x => ⟨↑x, ⋯⟩) ⟨(pbo f).inclusion x, ⋯⟩)).toCommRingCatIso.inv =
    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv
state_after: no goals
tactic: rfl