Module: Mathlib.RingTheory.Flat.Equalizer

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
T : Type u_3
inst✝⁸ : CommRing T
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S T
inst✝⁵ : IsScalarTower R S T
A : Type u_4
B : Type u_5
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra R A
inst✝¹ : Algebra R B
f g : A →ₐ[R] B
inst✝ : Module.Flat R T
h₁ : Function.Surjective ⇑(algebraMap S T)
φ : A ⊗[R] S →ₐ[A] A ⊗[R] T := TensorProduct.map (AlgHom.id A A) (IsScalarTower.toAlgHom R S T)
ePp : A ⊗[R] S ≃ₐ[S] S ⊗[R] A :=
  let __spread.0 := TensorProduct.comm R A S;
  { toEquiv := __spread.0.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }
e₃ : ↥(RingHom.ker φ) ≃ₗ[R] A ⊗[R] ↥(RingHom.ker (algebraMap S T)) :=
  LinearEquiv.restrictScalars R ((IsScalarTower.toAlgHom R S T).toLinearMap.kerLTensorEquivOfSurjective h₁ A)
e₄' : ↥(RingHom.ker φ) ≃ₗ[R] A ⊗[R] S ⊗[S] ↥(RingHom.ker (algebraMap S T)) :=
  e₃ ≪≫ₗ _root_.TensorProduct.comm R A ↥(RingHom.ker (algebraMap S T)) ≪≫ₗ
        LinearEquiv.restrictScalars R
          (AlgebraTensorModule.cancelBaseChange R S S (↥(RingHom.ker (algebraMap S T))) A).symm ≪≫ₗ
      LinearEquiv.restrictScalars R
        (AlgebraTensorModule.congr (LinearEquiv.refl S ↥(RingHom.ker (algebraMap S T))) ↑ePp.symm) ≪≫ₗ
    LinearEquiv.restrictScalars R (_root_.TensorProduct.comm S (↥(RingHom.ker (algebraMap S T))) (A ⊗[R] S))
⊢ ↥(RingHom.ker (TensorProduct.map (AlgHom.id A A) (IsScalarTower.toAlgHom R S T))) ≃ₗ[A ⊗[R] S]
    A ⊗[R] S ⊗[S] ↥(RingHom.ker (algebraMap S T))

Tactic:
let e₄ : (A ⊗[R] S) ⊗[S] (RingHom.ker (algebraMap S T)) ≃ₗ[A ⊗[R] S] (RingHom.ker φ) :=
    { __ := e₄'.symm, map_smul' r' x := by
        dsimp
        induction x with
        | zero => simp only [smul_zero, LinearEquiv.map_zero]
        | add x y _ _ => simp only [smul_add, LinearEquiv.map_add, *]
        | tmul x y =>
        induction x with
        | zero => simp only [zero_tmul, smul_zero, LinearEquiv.map_zero]
        | add x y _ _ => simp only [smul_add, add_tmul, LinearEquiv.map_add, *]
        | tmul x z =>
        induction r' with
        | zero => simp only [zero_smul, LinearEquiv.map_zero]
        | add x y _ _ => simp only [add_smul, LinearEquiv.map_add, *]
        | tmul r s =>
        rw [smul_tmul']
        ext1
        dsimp [e₄', ePp, φ]
        change ((r * x) ⊗ₜ[R] ((s * z) * y.1)) = (r ⊗ₜ[R] s) * (x ⊗ₜ[R] (z * y.1))
        rw [Algebra.TensorProduct.tmul_mul_tmul, mul_assoc] }

Goal state after:
R : Type u_1
S : Type u_2
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
T : Type u_3
inst✝⁸ : CommRing T
inst✝⁷ : Algebra R T
inst✝⁶ : Algebra S T
inst✝⁵ : IsScalarTower R S T
A : Type u_4
B : Type u_5
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra R A
inst✝¹ : Algebra R B
f g : A →ₐ[R] B
inst✝ : Module.Flat R T
h₁ : Function.Surjective ⇑(algebraMap S T)
φ : A ⊗[R] S →ₐ[A] A ⊗[R] T := TensorProduct.map (AlgHom.id A A) (IsScalarTower.toAlgHom R S T)
ePp : A ⊗[R] S ≃ₐ[S] S ⊗[R] A :=
  let __spread.0 := TensorProduct.comm R A S;
  { toEquiv := __spread.0.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }
e₃ : ↥(RingHom.ker φ) ≃ₗ[R] A ⊗[R] ↥(RingHom.ker (algebraMap S T)) :=
  LinearEquiv.restrictScalars R ((IsScalarTower.toAlgHom R S T).toLinearMap.kerLTensorEquivOfSurjective h₁ A)
e₄' : ↥(RingHom.ker φ) ≃ₗ[R] A ⊗[R] S ⊗[S] ↥(RingHom.ker (algebraMap S T)) :=
  e₃ ≪≫ₗ _root_.TensorProduct.comm R A ↥(RingHom.ker (algebraMap S T)) ≪≫ₗ
        LinearEquiv.restrictScalars R
          (AlgebraTensorModule.cancelBaseChange R S S (↥(RingHom.ker (algebraMap S T))) A).symm ≪≫ₗ
      LinearEquiv.restrictScalars R
        (AlgebraTensorModule.congr (LinearEquiv.refl S ↥(RingHom.ker (algebraMap S T))) ↑ePp.symm) ≪≫ₗ
    LinearEquiv.restrictScalars R (_root_.TensorProduct.comm S (↥(RingHom.ker (algebraMap S T))) (A ⊗[R] S))
e₄ : A ⊗[R] S ⊗[S] ↥(RingHom.ker (algebraMap S T)) ≃ₗ[A ⊗[R] S] ↥(RingHom.ker φ) :=
  let __spread.0 := e₄'.symm;
  { toAddHom := (↑__spread.0).toAddHom, map_smul' := ⋯, invFun := __spread.0.invFun, left_inv := ⋯, right_inv := ⋯ }
⊢ ↥(RingHom.ker (TensorProduct.map (AlgHom.id A A) (IsScalarTower.toAlgHom R S T))) ≃ₗ[A ⊗[R] S]
    A ⊗[R] S ⊗[S] ↥(RingHom.ker (algebraMap S T))