-- Module: Mathlib.RingTheory.Polynomial.UniversalFactorizationRing
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
n m k : ℕ
hn : n = m + k
p : MonicDegreeEq R n
q : { q : MonicDegreeEq S m × MonicDegreeEq S k // (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) }
this_1 : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=
  RingHom.toAlgebra (AlgHom.toRingHom (MvPolynomial.universalFactorizationMap R n m k hn))
this : Algebra (MvPolynomial (Fin n) R) R :=
  RingHom.toAlgebra
    (AlgHom.toRingHom (R := R)
      ((Equiv.symm (MvPolynomial.mapEquivMonic R R n) : (a : MonicDegreeEq R n) → MvPolynomial (Fin n) R →ₐ[R] R) p))
n_1 : ℕ
⊢ coeff
      (↑(↑(Subtype.mk (p := fun (q : MonicDegreeEq S m × MonicDegreeEq S k) ↦
                  (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]))
                  (MonicDegreeEq.map
                      (MonicDegreeEq.map (MonicDegreeEq.freeMonic R m)
                        (↑(AlgHom.comp (AlgHom.restrictScalars R Algebra.TensorProduct.includeRight)
                              Algebra.TensorProduct.includeLeft) :
                          MvPolynomial (Fin m) R →+*
                            R ⊗[MvPolynomial (Fin n) R] (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)))
                      (↑(Algebra.TensorProduct.lift (Algebra.ofId R S)
                              {
                                toRingHom :=
                                  (↑(Algebra.TensorProduct.lift
                                          (MvPolynomial.aeval fun (x : Fin m) ↦
                                            (↑(↑q : MonicDegreeEq S m × MonicDegreeEq S k).1 : S[X]).coeff (↑x : ℕ))
                                          (MvPolynomial.aeval fun (x : Fin k) ↦
                                            (↑(↑q : MonicDegreeEq S m × MonicDegreeEq S k).2 : S[X]).coeff (↑x : ℕ))
                                          ⋯ :
                                        MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S) :
                                    MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →+* S),
                                commutes' := ⋯ :
                                AlgHom.{u_1, u_1, u_2} (MvPolynomial (Fin n) R)
                                  (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) S }
                              ⋯ :
                            R ⊗[MvPolynomial (Fin n) R] (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) →ₐ[R] S) :
                        R ⊗[MvPolynomial (Fin n) R] (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) →+* S),
                    MonicDegreeEq.map
                      (MonicDegreeEq.map (MonicDegreeEq.freeMonic R k)
                        (↑(AlgHom.comp (AlgHom.restrictScalars R Algebra.TensorProduct.includeRight)
                              Algebra.TensorProduct.includeRight) :
                          MvPolynomial (Fin k) R →+*
                            R ⊗[MvPolynomial (Fin n) R] (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)))
                      (↑(Algebra.TensorProduct.lift (Algebra.ofId R S)
                              {
                                toRingHom :=
                                  (↑(Algebra.TensorProduct.lift
                                          (MvPolynomial.aeval fun (x : Fin m) ↦
                                            (↑(↑q : MonicDegreeEq S m × MonicDegreeEq S k).1 : S[X]).coeff (↑x : ℕ))
                                          (MvPolynomial.aeval fun (x : Fin k) ↦
                                            (↑(↑q : MonicDegreeEq S m × MonicDegreeEq S k).2 : S[X]).coeff (↑x : ℕ))
                                          ⋯ :
                                        MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S) :
                                    MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →+* S),
                                commutes' := ⋯ :
                                AlgHom.{u_1, u_1, u_2} (MvPolynomial (Fin n) R)
                                  (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) S }
                              ⋯ :
                            R ⊗[MvPolynomial (Fin n) R] (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) →ₐ[R] S) :
                        R ⊗[MvPolynomial (Fin n) R] (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) →+* S))
                  ⋯) :
              MonicDegreeEq S m × MonicDegreeEq S k).1 :
        S[X])
      n_1 =
    (↑(↑q : MonicDegreeEq S m × MonicDegreeEq S k).1 : S[X]).coeff n_1

Tactic:
simp +contextual [coeff_freeMonic, apply_dite, MonicDegreeEq.coeff_of_ge]

Premises used: apply_dite, Polynomial.MonicDegreeEq.coeff_of_ge, Polynomial.coeff_freeMonic