-- Module: Mathlib.RingTheory.Spectrum.Prime.FreeLocus
Goal:
R : Type uR
M : Type uM
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : FinitePresentation R M
x_1 : (↑(freeLocus R M) : Type uR)
x : PrimeSpectrum R
hx : Membership.mem (γ := Set (PrimeSpectrum R)) (freeLocus R M) x
this_1 : Free (Localization.AtPrime x.asIdeal) (LocalizedModule x.asIdeal.primeCompl M)
f : R
hf : f ∈ x.asIdeal.primeCompl
hf' : Free (Localization (Submonoid.powers f)) (LocalizedModule.Away f M)
hf'' :
  finrank (Localization (Submonoid.powers f)) (LocalizedModule.Away f M) =
    finrank (Localization.AtPrime x.asIdeal) (LocalizedModule x.asIdeal.primeCompl M)
p : PrimeSpectrum R
hp'' : Membership.mem (γ := Set (PrimeSpectrum R)) (freeLocus R M) p
hp :
  Membership.mem (γ := Set (↑(freeLocus R M) : Type uR))
    (Subtype.val (p := fun (x : PrimeSpectrum R) ↦ Membership.mem (γ := Set (PrimeSpectrum R)) (freeLocus R M) x) ⁻¹'
      (↑(basicOpen f) : Set (PrimeSpectrum R)))
    (Subtype.mk (p := fun (x : PrimeSpectrum R) ↦ Membership.mem (γ := Set (PrimeSpectrum R)) (freeLocus R M) x) p hp'')
p' : Submonoid (Localization (Submonoid.powers f)) :=
  Algebra.algebraMapSubmonoid (Localization (Submonoid.powers f)) p.asIdeal.primeCompl
hp' : Submonoid.powers f ≤ p.asIdeal.primeCompl
Rₚ : Type uR := Localization.AtPrime p.asIdeal
Mₚ : Type (max uR uM) := LocalizedModule p.asIdeal.primeCompl M
this_2 : Algebra (Localization.Away f) Rₚ :=
  IsLocalization.localizationAlgebraOfSubmonoidLe (Localization.Away f) Rₚ (Submonoid.powers f) p.asIdeal.primeCompl hp'
this_3 : IsScalarTower R (Localization.Away f) Rₚ
this_4 : Module (Localization.Away f) Mₚ := compHom Mₚ (algebraMap (Localization.Away f) Rₚ)
this_5 : IsScalarTower R (Localization.Away f) Mₚ
this_6 : IsScalarTower (Localization.Away f) Rₚ Mₚ
l : LinearMap (R := Localization.Away f) (RingHom.id (Localization.Away f)) (LocalizedModule (Submonoid.powers f) M)
  (LocalizedModule p.asIdeal.primeCompl M) :=
  LinearMap.extendScalarsOfIsLocalization (Submonoid.powers f) (Localization.Away f)
    (IsLocalizedModule.liftOfLE (Submonoid.powers f) p.asIdeal.primeCompl hp'
      (LocalizedModule.mkLinearMap (Submonoid.powers f) M) (LocalizedModule.mkLinearMap p.asIdeal.primeCompl M))
this_7 : IsLocalization p' Rₚ
this_8 :
  IsLocalizedModule p.asIdeal.primeCompl
    ((↑R :
        {S : Type uR} →
          {M M₂ : Type (max uM uR)} →
            [inst : Semiring R] →
              [inst_4 : Semiring S] →
                [inst_5 : AddCommMonoid M] →
                  [inst_6 : AddCommMonoid M₂] →
                    [inst_7 : Module R M] →
                      [inst_8 : Module R M₂] →
                        [inst_9 : Module S M] →
                          [inst_10 : Module S M₂] →
                            [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
      l)
this_9 : IsLocalizedModule (Algebra.algebraMapSubmonoid (Localization.Away f) p.asIdeal.primeCompl) l
this :
  finrank Rₚ (LocalizedModule p.asIdeal.primeCompl M) =
    finrank (Localization (Submonoid.powers f)) (LocalizedModule (Submonoid.powers f) M)
⊢ rankAtStalk M
      (↑(Subtype.mk (p := fun (x : PrimeSpectrum R) ↦ Membership.mem (γ := Set (PrimeSpectrum R)) (freeLocus R M) x) p
            hp'') :
        PrimeSpectrum R) =
    rankAtStalk M
      (↑(Subtype.mk (p := fun (x : PrimeSpectrum R) ↦ Membership.mem (γ := Set (PrimeSpectrum R)) (freeLocus R M) x) x
            hx) :
        PrimeSpectrum R)

Tactic:
simp [Rₚ, rankAtStalk, this, hf'']

Premises used: Module.rankAtStalk