Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: case refine_1
R : CommRingCat
M : (Spec R).Modules
φ : (f : ↑R) →
  ↑((modulesSpecToSheaf.obj M).obj.obj (op ⊤)) →ₗ[↑R] ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f))) :=
  fun f ↦ ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)
h : IsLocalizing (modulesSpecToSheaf.obj M)
⊢ Aux M ⊤

pp: case refine_2
R : CommRingCat
M : (Spec R).Modules
φ : (f : ↑R) →
  ↑((modulesSpecToSheaf.obj M).obj.obj (op ⊤)) →ₗ[↑R] ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f))) :=
  fun f ↦ ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)
h : Aux M ⊤
f : ↑R
⊢ IsUnit ((algebraMap (↑R) (Module.End ↑R ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f))))) f)

pp: case refine_3
R : CommRingCat
M : (Spec R).Modules
φ : (f : ↑R) →
  ↑((modulesSpecToSheaf.obj M).obj.obj (op ⊤)) →ₗ[↑R] ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f))) :=
  fun f ↦ ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)
h : Aux M ⊤
f : ↑R
⊢ ∀ (x : ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f)))),
    ∃ n y, f ^ n • x = (ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)) y

pp: case refine_4
R : CommRingCat
M : (Spec R).Modules
φ : (f : ↑R) →
  ↑((modulesSpecToSheaf.obj M).obj.obj (op ⊤)) →ₗ[↑R] ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f))) :=
  fun f ↦ ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)
h : Aux M ⊤
f : ↑R
⊢ ∀ (x : ↑((modulesSpecToSheaf.obj M).obj.obj (op ⊤))),
    (ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)) x = 0 → ∃ n, f ^ n • x = 0

Tactic:
· have hf (f : R) : IsLocalizedModule.Away f (φ f) := h f
    refine ⟨fun f hle s ↦ ?_, fun f hle s hs ↦ ?_⟩
    · obtain ⟨n, y, hy⟩ := (hf f).surj _ _ s
      use n, y, hy.symm
    · obtain ⟨⟨_, n, rfl⟩, hn⟩ := (IsLocalizedModule.eq_zero_iff (.powers f) (φ f)).mp hs
      use n, hn

Goal state after:
case refine_2
R : CommRingCat
M : (Spec R).Modules
φ : (f : ↑R) →
  ↑((modulesSpecToSheaf.obj M).obj.obj (op ⊤)) →ₗ[↑R] ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f))) :=
  fun f ↦ ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)
h : Aux M ⊤
f : ↑R
⊢ IsUnit ((algebraMap (↑R) (Module.End ↑R ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f))))) f)

case refine_3
R : CommRingCat
M : (Spec R).Modules
φ : (f : ↑R) →
  ↑((modulesSpecToSheaf.obj M).obj.obj (op ⊤)) →ₗ[↑R] ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f))) :=
  fun f ↦ ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)
h : Aux M ⊤
f : ↑R
⊢ ∀ (x : ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f)))),
    ∃ n y, f ^ n • x = (ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)) y

case refine_4
R : CommRingCat
M : (Spec R).Modules
φ : (f : ↑R) →
  ↑((modulesSpecToSheaf.obj M).obj.obj (op ⊤)) →ₗ[↑R] ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f))) :=
  fun f ↦ ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)
h : Aux M ⊤
f : ↑R
⊢ ∀ (x : ↑((modulesSpecToSheaf.obj M).obj.obj (op ⊤))),
    (ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)) x = 0 → ∃ n, f ^ n • x = 0