Module: Mathlib.RingTheory.Flat.Equalizer

Goal state before:
pp: case tmul.tmul.tmul
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 := ⋯
ePp : A ⊗[R] S ≃ₐ[S] S ⊗[R] A := ⋯
e₃ : ↥(RingHom.ker φ) ≃ₗ[R] A ⊗[R] ↥(RingHom.ker (algebraMap S T)) := ⋯
e₄' : ↥(RingHom.ker φ) ≃ₗ[R] A ⊗[R] S ⊗[S] ↥(RingHom.ker (algebraMap S T)) := ⋯
y : ↥(RingHom.ker (algebraMap S T))
x : A
z : S
r : A
s : S
⊢ ↑(e₄'.symm ((r ⊗ₜ[R] s • x ⊗ₜ[R] z) ⊗ₜ[S] y)) = ↑(r ⊗ₜ[R] s • e₄'.symm (x ⊗ₜ[R] z ⊗ₜ[S] y))

Tactic:
dsimp [e₄', ePp, φ]

Goal state after:
case tmul.tmul.tmul
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))
y : ↥(RingHom.ker (algebraMap S T))
x : A
z : S
r : A
s : S
⊢ ↑(e₃.symm
        ((_root_.TensorProduct.comm R (↥(RingHom.ker (algebraMap S T))) A)
          ((LinearEquiv.restrictScalars R
                (AlgebraTensorModule.cancelBaseChange R S S (↥(RingHom.ker (algebraMap S T))) A).symm).symm
            ((LinearEquiv.restrictScalars R
                  (AlgebraTensorModule.congr (LinearEquiv.refl S ↥(RingHom.ker (algebraMap S T)))
                    (↑{ toEquiv := ↑(TensorProduct.comm R A S), map_mul' := ⋯, map_add' := ⋯,
                          commutes' := ⋯ }).symm)).symm
              ((LinearEquiv.restrictScalars R
                    (_root_.TensorProduct.comm S (↥(RingHom.ker (algebraMap S T))) (A ⊗[R] S))).symm
                ((r * x) ⊗ₜ[R] (s * z) ⊗ₜ[S] y)))))) =
    r ⊗ₜ[R] s *
      ↑(e₃.symm
          ((_root_.TensorProduct.comm R (↥(RingHom.ker (algebraMap S T))) A)
            ((LinearEquiv.restrictScalars R
                  (AlgebraTensorModule.cancelBaseChange R S S (↥(RingHom.ker (algebraMap S T))) A).symm).symm
              ((LinearEquiv.restrictScalars R
                    (AlgebraTensorModule.congr (LinearEquiv.refl S ↥(RingHom.ker (algebraMap S T)))
                      (↑{ toEquiv := ↑(TensorProduct.comm R A S), map_mul' := ⋯, map_add' := ⋯,
                            commutes' := ⋯ }).symm)).symm
                ((LinearEquiv.restrictScalars R
                      (_root_.TensorProduct.comm S (↥(RingHom.ker (algebraMap S T))) (A ⊗[R] S))).symm
                  (x ⊗ₜ[R] z ⊗ₜ[S] y))))))