-- Module: Mathlib.RingTheory.LocalProperties.Projective
Goal:
R : Type u_1
N : Type u_2
M : Type uM
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
f : M →ₗ[R] N
inst_5 : Module.FinitePresentation R N
H :
  ∀ (I : Ideal R) (x : I.IsMaximal),
    ∃ (g :
      LinearMap (R := Localization.AtPrime I) (RingHom.id (Localization.AtPrime I)) (LocalizedModule I.primeCompl N)
        (LocalizedModule I.primeCompl M)),
      (LocalizedModule.map I.primeCompl :
              (a : M →ₗ[R] N) →
                LocalizedModule I.primeCompl M →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N)
            f ∘ₗ
          g =
        id
I : Ideal R
hI : I.IsMaximal
this :
  Membership.mem (γ :=
    Submodule (Localization (M := R) I.primeCompl)
      (LinearMap (R := Localization (M := R) I.primeCompl) (RingHom.id (Localization (M := R) I.primeCompl))
        (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl N)))
    (range (R := Localization (M := R) I.primeCompl) (M :=
      LinearMap (R := Localization (M := R) I.primeCompl) (RingHom.id (Localization.AtPrime I))
        (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M))
      (τ₁₂ := RingHom.id (Localization (M := R) I.primeCompl))
      ((llcomp (Localization (M := R) I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
            (LocalizedModule I.primeCompl N) :
          (a : LocalizedModule I.primeCompl M →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N) →
            (LocalizedModule I.primeCompl N →ₗ[Localization.AtPrime I]
                LocalizedModule I.primeCompl M) →ₗ[Localization I.primeCompl]
              LocalizedModule I.primeCompl N →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N)
        ((LocalizedModule.map I.primeCompl :
            (a : M →ₗ[R] N) →
              LocalizedModule I.primeCompl M →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N)
          f)))
    id
c : ↥I.primeCompl
g : N →ₗ[R] M
x : N
⊢ Eq (α := LocalizedModule I.primeCompl N)
    (((↑R :
                {S : Type u_1} →
                  {M M₂ : Type (max u_1 u_2)} →
                    [inst : Semiring R] →
                      [inst_6 : Semiring S] →
                        [inst_7 : AddCommMonoid M] →
                          [inst_8 : AddCommMonoid M₂] →
                            [inst_9 : Module R M] →
                              [inst_10 : Module R M₂] →
                                [inst_11 : Module S M] →
                                  [inst_12 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
              (S := Localization (M := R) I.primeCompl)
              ((((llcomp (Localization (M := R) I.primeCompl) (LocalizedModule I.primeCompl N)
                          (LocalizedModule I.primeCompl M) (LocalizedModule I.primeCompl N) :
                        (a :
                            LocalizedModule I.primeCompl M →ₗ[Localization I.primeCompl]
                              LocalizedModule I.primeCompl N) →
                          (LocalizedModule I.primeCompl N →ₗ[Localization.AtPrime I]
                              LocalizedModule I.primeCompl M) →ₗ[Localization I.primeCompl]
                            LocalizedModule I.primeCompl N →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N)
                      ((LocalizedModule.map I.primeCompl :
                          (a : M →ₗ[R] N) →
                            LocalizedModule I.primeCompl M →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N)
                        f) :
                    LinearMap (R := Localization (M := R) I.primeCompl)
                      (RingHom.id (Localization (M := R) I.primeCompl))
                      (LinearMap (R := Localization (M := R) I.primeCompl) (RingHom.id (Localization.AtPrime I))
                        (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M))
                      (LinearMap (R := Localization (M := R) I.primeCompl)
                        (RingHom.id (Localization (M := R) I.primeCompl)) (LocalizedModule I.primeCompl N)
                        (LocalizedModule I.primeCompl N))) :
                  (a : LocalizedModule I.primeCompl N →ₗ[Localization.AtPrime I] LocalizedModule I.primeCompl M) →
                    LocalizedModule I.primeCompl N →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N)
                ((LocalizedModule.map I.primeCompl :
                    (a : N →ₗ[R] M) →
                      LocalizedModule I.primeCompl N →ₗ[Localization.AtPrime I] LocalizedModule I.primeCompl M)
                  g)) ∘ₗ
            LocalizedModule.mkLinearMap I.primeCompl N :
          (a : N) → LocalizedModule I.primeCompl N)
        x :
      LocalizedModule I.primeCompl N)
    (((↑R :
                {S : Type u_1} →
                  {M M₂ : Type (max u_1 u_2)} →
                    [inst : Semiring R] →
                      [inst_6 : Semiring S] →
                        [inst_7 : AddCommMonoid M] →
                          [inst_8 : AddCommMonoid M₂] →
                            [inst_9 : Module R M] →
                              [inst_10 : Module R M₂] →
                                [inst_11 : Module S M] →
                                  [inst_12 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
              (S := Localization (M := R) I.primeCompl)
              ((LocalizedModule.map I.primeCompl :
                  (a : N →ₗ[R] N) →
                    LocalizedModule I.primeCompl N →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N)
                ((((llcomp R N M N : (a : M →ₗ[R] N) → (N →ₗ[R] M) →ₗ[R] N →ₗ[R] N) f : (N →ₗ[R] M) →ₗ[R] N →ₗ[R] N) :
                    (a : N →ₗ[R] M) → N →ₗ[R] N)
                  g)) ∘ₗ
            LocalizedModule.mkLinearMap I.primeCompl N :
          (a : N) → LocalizedModule I.primeCompl N)
        x :
      LocalizedModule I.primeCompl N)

Tactic:
simp only [LocalizedModule.map_mk, LinearMap.coe_comp, LinearMap.coe_restrictScalars,
        Function.comp_apply, LocalizedModule.mkLinearMap_apply, LinearMap.llcomp_apply,
        LocalizedModule.map_mk]

Premises used: LinearMap.llcomp_apply, LinearMap.coe_restrictScalars, LinearMap.coe_comp, Function.comp_apply, LocalizedModule.map_mk, LocalizedModule.mkLinearMap_apply