-- Module: Mathlib.RingTheory.Localization.Free
Goal:
R : Type u_4
M : Type u_5
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
S : Submonoid R
M' : Type u_1
inst_3 : AddCommGroup M'
inst_4 : Module R M'
f : M →ₗ[R] M'
inst_5 : IsLocalizedModule S f
Rₛ : Type u_3
inst_6 : CommRing Rₛ
inst_7 : Algebra R Rₛ
inst_8 : Module Rₛ M'
inst_9 : IsScalarTower R Rₛ M'
inst_10 : IsLocalization S Rₛ
inst_11 : FinitePresentation R M
I : Type u_6
inst_12 : Finite I
b : Basis I Rₛ M'
this_1 : FinitePresentation R (I →₀ R)
r : R
hr : r ∈ S
e : LocalizedModule.Away r M ≃ₗ[Localization (Submonoid.powers r)] LocalizedModule.Away r (I →₀ R)
e' : LocalizedModule (Submonoid.powers r) (I →₀ R) ≃ₗ[R] Finsupp.{u_6, u_4} I (Localization (Submonoid.powers r)) :=
  IsLocalizedModule.iso (Submonoid.powers r)
    (Finsupp.mapRange.linearMap (Algebra.linearMap R (Localization (Submonoid.powers r))))
i : I
this :
  Eq (α := LocalizedModule (Submonoid.powers r) (I →₀ R))
    ((e'.symm : (a : I →₀ Localization (Submonoid.powers r)) → LocalizedModule (Submonoid.powers r) (I →₀ R))
        (Finsupp.single i 1) :
      LocalizedModule (Submonoid.powers r) (I →₀ R))
    (LocalizedModule.mk (Finsupp.single i 1) 1 : LocalizedModule (Submonoid.powers r) (I →₀ R))
he :
  Eq (α := I →₀ Rₛ)
    ((LocalizedModule.lift (Submonoid.powers r) (Finsupp.mapRange.linearMap (Algebra.linearMap R Rₛ)) ⋯ ∘ₗ
            (↑R :
                {S : Type u_4} →
                  {M : Type (max u_4 u_5)} →
                    {M₂ : Type (max u_4 u_6)} →
                      [inst : Semiring R] →
                        [inst_13 : Semiring S] →
                          [inst_14 : AddCommMonoid M] →
                            [inst_15 : AddCommMonoid M₂] →
                              [inst_16 : Module R M] →
                                [inst_17 : Module R M₂] →
                                  [inst_18 : Module S M] →
                                    [inst_19 : Module S M₂] →
                                      [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
              (↑e : LocalizedModule.Away r M →ₗ[Localization (Submonoid.powers r)] LocalizedModule.Away r (I →₀ R)) :
          (a : LocalizedModule (Submonoid.powers r) M) → I →₀ Rₛ)
        ((e.symm : (a : LocalizedModule.Away r (I →₀ R)) → LocalizedModule.Away r M)
          ((e'.symm : (a : I →₀ Localization (Submonoid.powers r)) → LocalizedModule (Submonoid.powers r) (I →₀ R))
            (Finsupp.single i 1))) :
      I →₀ Rₛ)
    (((↑(LinearEquiv.restrictScalars R b.repr) : M' →ₗ[R] I →₀ Rₛ) ∘ₗ LocalizedModule.lift (Submonoid.powers r) f ⋯ :
          (a : LocalizedModule (Submonoid.powers r) M) → I →₀ Rₛ)
        ((e.symm : (a : LocalizedModule.Away r (I →₀ R)) → LocalizedModule.Away r M)
          ((e'.symm : (a : I →₀ Localization (Submonoid.powers r)) → LocalizedModule (Submonoid.powers r) (I →₀ R))
            (Finsupp.single i 1))) :
      I →₀ Rₛ)
⊢ Eq (α := I →₀ Rₛ)
    ((LocalizedModule.lift (Submonoid.powers r) (Finsupp.mapRange.linearMap (Algebra.linearMap R Rₛ)) ⋯ :
          (a : LocalizedModule (Submonoid.powers r) (I →₀ R)) → I →₀ Rₛ)
        ((e'.symm : (a : I →₀ Localization (Submonoid.powers r)) → LocalizedModule (Submonoid.powers r) (I →₀ R))
          (Finsupp.single i 1)) :
      I →₀ Rₛ)
    ((b.repr : (a : M') → I →₀ Rₛ)
        ((LocalizedModule.lift (Submonoid.powers r) f ⋯ : (a : LocalizedModule (Submonoid.powers r) M) → M')
          ((e.symm : (a : LocalizedModule.Away r (I →₀ R)) → LocalizedModule.Away r M)
              ((e'.symm : (a : I →₀ Localization (Submonoid.powers r)) → LocalizedModule (Submonoid.powers r) (I →₀ R))
                (Finsupp.single i 1)) :
            LocalizedModule.Away r M)) :
      I →₀ Rₛ)

Tactic:
simp only [LinearMap.coe_comp, LinearMap.coe_restrictScalars, LinearEquiv.coe_coe,
      Function.comp_apply, LinearEquiv.apply_symm_apply, LinearEquiv.restrictScalars_apply] at he

Premises used: LinearEquiv.restrictScalars_apply, LinearMap.coe_restrictScalars, LinearMap.coe_comp, LinearEquiv.apply_symm_apply, Function.comp_apply, LinearEquiv.coe_coe