Module: Mathlib.Algebra.Module.FinitePresentation

Goal state before:
pp: R : Type u_3
M : Type u_4
N : Type u_5
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
S : Submonoid R
M' : Type u_1
inst✝⁷ : AddCommGroup M'
inst✝⁶ : Module R M'
f : M →ₗ[R] M'
inst✝⁵ : IsLocalizedModule S f
N' : Type u_2
inst✝⁴ : AddCommGroup N'
inst✝³ : Module R N'
g : N →ₗ[R] N'
inst✝² : IsLocalizedModule S g
inst✝¹ : Module.Finite R M
inst✝ : Module.FinitePresentation R N
l : M →ₗ[R] N
hf : Function.Bijective ⇑((IsLocalizedModule.map S f g) l)
e : M' ≃ₗ[R] N' := ⋯
l' : N →ₗ[R] M
s₀ : ↥S
H : f ∘ₗ l' = s₀ • ↑e.symm ∘ₗ g
H₁ : g ∘ₗ l ∘ₗ l' = g ∘ₗ (s₀ • LinearMap.id)
s₁ : ↥S
hs₁ : s₁ • l ∘ₗ l' = s₁ • s₀ • LinearMap.id
H₂ : f ∘ₗ l' ∘ₗ l = f ∘ₗ (s₀ • LinearMap.id)
s₂ : ↥S
hs₂ : s₂ • l' ∘ₗ l = s₂ • s₀ • LinearMap.id
t : R
ht : ↑s₀ * ↑s₁ * ↑s₂ ∣ t
Rₛ : Type u_3 := ⋯
lₛ : LocalizedModule (Submonoid.powers t) M →ₗ[Localization (Submonoid.powers t)]
  LocalizedModule (Submonoid.powers t) N := ⋯
hu : IsUnit ((algebraMap R Rₛ) ↑⟨t, ⋯⟩)
hu₀ : IsUnit ((algebraMap R Rₛ) ↑s₀)
hu₁ : IsUnit ((algebraMap R Rₛ) ↑s₁)
hu₂ : IsUnit ((algebraMap R Rₛ) ↑s₂)
lₛ' : LocalizedModule (Submonoid.powers t) N →ₗ[Localization (Submonoid.powers t)]
  LocalizedModule (Submonoid.powers t) M := ⋯
H_left : (↑hu₀.unit⁻¹ • lₛ') ∘ₗ lₛ = LinearMap.id
⊢ ((algebraMap Rₛ
          (Module.End Rₛ
            (LocalizedModule (Submonoid.powers t) N →ₗ[Localization (Submonoid.powers t)]
              LocalizedModule (Submonoid.powers t) N)))
        ((algebraMap R Rₛ) ↑s₁))
      (lₛ ∘ₗ (↑hu₀.unit⁻¹ • lₛ')) =
    ((algebraMap Rₛ
          (Module.End Rₛ
            (LocalizedModule (Submonoid.powers t) N →ₗ[Localization (Submonoid.powers t)]
              LocalizedModule (Submonoid.powers t) N)))
        ((algebraMap R Rₛ) ↑s₁))
      LinearMap.id

Tactic:
apply ((Module.End.isUnit_iff _).mp (hu₀.map (algebraMap Rₛ (Module.End Rₛ _)))).1

Goal state after:
R : Type u_3
M : Type u_4
N : Type u_5
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
S : Submonoid R
M' : Type u_1
inst✝⁷ : AddCommGroup M'
inst✝⁶ : Module R M'
f : M →ₗ[R] M'
inst✝⁵ : IsLocalizedModule S f
N' : Type u_2
inst✝⁴ : AddCommGroup N'
inst✝³ : Module R N'
g : N →ₗ[R] N'
inst✝² : IsLocalizedModule S g
inst✝¹ : Module.Finite R M
inst✝ : Module.FinitePresentation R N
l : M →ₗ[R] N
hf : Function.Bijective ⇑((IsLocalizedModule.map S f g) l)
e : M' ≃ₗ[R] N' := ⋯
l' : N →ₗ[R] M
s₀ : ↥S
H : f ∘ₗ l' = s₀ • ↑e.symm ∘ₗ g
H₁ : g ∘ₗ l ∘ₗ l' = g ∘ₗ (s₀ • LinearMap.id)
s₁ : ↥S
hs₁ : s₁ • l ∘ₗ l' = s₁ • s₀ • LinearMap.id
H₂ : f ∘ₗ l' ∘ₗ l = f ∘ₗ (s₀ • LinearMap.id)
s₂ : ↥S
hs₂ : s₂ • l' ∘ₗ l = s₂ • s₀ • LinearMap.id
t : R
ht : ↑s₀ * ↑s₁ * ↑s₂ ∣ t
Rₛ : Type u_3 := ⋯
lₛ : LocalizedModule (Submonoid.powers t) M →ₗ[Localization (Submonoid.powers t)]
  LocalizedModule (Submonoid.powers t) N := ⋯
hu : IsUnit ((algebraMap R Rₛ) ↑⟨t, ⋯⟩)
hu₀ : IsUnit ((algebraMap R Rₛ) ↑s₀)
hu₁ : IsUnit ((algebraMap R Rₛ) ↑s₁)
hu₂ : IsUnit ((algebraMap R Rₛ) ↑s₂)
lₛ' : LocalizedModule (Submonoid.powers t) N →ₗ[Localization (Submonoid.powers t)]
  LocalizedModule (Submonoid.powers t) M := ⋯
H_left : (↑hu₀.unit⁻¹ • lₛ') ∘ₗ lₛ = LinearMap.id
⊢ ((algebraMap Rₛ
          (Module.End Rₛ
            (LocalizedModule (Submonoid.powers t) N →ₗ[Localization (Submonoid.powers t)]
              LocalizedModule (Submonoid.powers t) N)))
        ((algebraMap R Rₛ) ↑s₀))
      (((algebraMap Rₛ
            (Module.End Rₛ
              (LocalizedModule (Submonoid.powers t) N →ₗ[Localization (Submonoid.powers t)]
                LocalizedModule (Submonoid.powers t) N)))
          ((algebraMap R Rₛ) ↑s₁))
        (lₛ ∘ₗ (↑hu₀.unit⁻¹ • lₛ'))) =
    ((algebraMap Rₛ
          (Module.End Rₛ
            (LocalizedModule (Submonoid.powers t) N →ₗ[Localization (Submonoid.powers t)]
              LocalizedModule (Submonoid.powers t) N)))
        ((algebraMap R Rₛ) ↑s₀))
      (((algebraMap Rₛ
            (Module.End Rₛ
              (LocalizedModule (Submonoid.powers t) N →ₗ[Localization (Submonoid.powers t)]
                LocalizedModule (Submonoid.powers t) N)))
          ((algebraMap R Rₛ) ↑s₁))
        LinearMap.id)