-- Lean 4 proof step from Mathlib.RingTheory.Extension.Cotangent.BaseChange (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
P : Extension.{u, u_1, u_2} R S
T : Type u_3
inst_3 : CommRing T
inst_4 : Algebra R T
inst_5 : Module.Flat R T
hc₁ : Eq (α := Unit →ₗ[R] Unit) (0 ∘ₗ 0) (0 ∘ₗ 0)
hc₂ :
  Eq (α := Unit →ₗ[R] Extension.H1Cotangent (R := T) (S := T ⊗[R] S) P.baseChange) (0 ∘ₗ 0)
    ((↑R :
          {S : Type u_3} →
            {M M₂ : Type (max u u_3)} →
              [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] →
                              [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
        (P.tensorToH1Cotangent T) ∘ₗ
      0)
⊢ Function.Bijective (α := TensorProduct.{u_1, u_3, u} R T P.H1Cotangent) (β :=
    Extension.H1Cotangent (R := T) (S := T ⊗[R] S) P.baseChange)
    (⇑(P.tensorToH1Cotangent T) : (a : T ⊗[R] P.H1Cotangent) → P.baseChange.H1Cotangent)
have hc₃ : Eq (α := TensorProduct.{u_1, u_3, u} R T P.H1Cotangent →ₗ[R] Cotangent (R := T) (S := T ⊗[R] S) P.baseChange)
  ((↑R :
        {S : Type (max u_2 u_3)} →
          {M M₂ : Type (max u u_3)} →
            [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] →
                            [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
      h1Cotangentι ∘ₗ
    (↑R :
        {S : Type u_3} →
          {M M₂ : Type (max u u_3)} →
            [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] →
                            [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
      (P.tensorToH1Cotangent T))
  ((↑(LinearEquiv.restrictScalars R (P.tensorCotangentOfFlat T)) : T ⊗[R] P.Cotangent →ₗ[R] P.baseChange.Cotangent) ∘ₗ
    LinearMap.lTensor T
      ((↑R :
          {S : Type u_2} →
            {M M₂ : Type u} →
              [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] →
                              [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
        h1Cotangentι))
Tactic: apply LinearMap.bijective_of_surjective_of_bijective_of_bijective_of_injective (M₁ := Unit)
      (N₁ := Unit) (M₂ := Unit) (N₂ := Unit)
    -- The row `0 → 0 → T ⊗ H¹(P) → T ⊗ P.Cotangent → T ⊗ P.CotangentSpace`.
    0 0
    ((P.h1Cotangentι.restrictScalars R).lTensor T)
    ((P.cotangentComplex.restrictScalars R).lTensor T)
    -- The row `0 → 0 → H¹(T ⊗ P) → (T ⊗ P).Cotangent → (T ⊗ P).CotangentSpace`.
    0 0
    (h1Cotangentι.restrictScalars R)
    ((P.baseChange (T := T)).cotangentComplex.restrictScalars R)
    -- The vertical maps induced by base change.
    0 0
    ((P.tensorToH1Cotangent T).restrictScalars R)
    ((P.tensorCotangentOfFlat T).restrictScalars R)
    ((P.tensorCotangentSpace T).restrictScalars R)