-- Module: Mathlib.RingTheory.Extension.Presentation.Core
Goal:
R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
P : Presentation R S ι σ
R₀ : Type u_5
inst_3 : CommRing R₀
inst_4 : Algebra R₀ R
inst_5 : Algebra R₀ S
inst_6 : IsScalarTower R₀ R S
inst_7 : P.HasCoeffs R₀
rw :
  Set.range P.relation ⊆
    (↑(RingHom.ker (F :=
          MvPolynomial ι R →ₐ[R]
            R ⊗[R₀]
              HasQuotient.Quotient (MvPolynomial ι R₀) (B := Ideal (MvPolynomial ι R₀))
                (Ideal.span
                  (Set.range (ι := σ) (relationOfHasCoeffs.{u_1, u_2, u_3, u_4, u_5} (R := R) (S := S) (P := P) R₀))))
          (AlgHom.comp (B := R ⊗[R₀] MvPolynomial ι R₀)
              (TensorProduct.map (AlgHom.id R R)
                  (Ideal.Quotient.mkₐ R₀
                    (Ideal.span
                      (Set.range (ι := σ)
                        (relationOfHasCoeffs.{u_1, u_2, u_3, u_4, u_5} (R := R) (S := S) (P := P) R₀)))) :
                R ⊗[R₀] MvPolynomial ι R₀ →ₐ[R]
                  R ⊗[R₀]
                    HasQuotient.Quotient (MvPolynomial ι R₀) (B := Ideal (MvPolynomial ι R₀))
                      (Ideal.span
                        (Set.range (ι := σ)
                          (relationOfHasCoeffs.{u_1, u_2, u_3, u_4, u_5} (R := R) (S := S) (P := P) R₀))))
              ((↑(AlgEquiv.symm (MvPolynomial.algebraTensorAlgEquiv R₀ R)) :
                  MvPolynomial ι R →ₐ[R] R ⊗[R₀] MvPolynomial ι R₀) :
                MvPolynomial ι R →ₐ[R] R ⊗[R₀] MvPolynomial ι R₀) :
            MvPolynomial ι R →ₐ[R]
              R ⊗[R₀]
                HasQuotient.Quotient (MvPolynomial ι R₀) (B := Ideal (MvPolynomial ι R₀))
                  (Ideal.span
                    (Set.range (ι := σ)
                      (relationOfHasCoeffs.{u_1, u_2, u_3, u_4, u_5} (R := R) (S := S) (P := P) R₀))))) :
      Set P.Ring)
⊢ P.ker ≤
    RingHom.ker (F :=
      MvPolynomial ι R →ₐ[R]
        R ⊗[R₀]
          HasQuotient.Quotient (MvPolynomial ι R₀) (B := Ideal (MvPolynomial ι R₀))
            (Ideal.span
              (Set.range (ι := σ) (relationOfHasCoeffs.{u_1, u_2, u_3, u_4, u_5} (R := R) (S := S) (P := P) R₀))))
      (AlgHom.comp (B := R ⊗[R₀] MvPolynomial ι R₀)
          (TensorProduct.map (AlgHom.id R R)
              (Ideal.Quotient.mkₐ R₀
                (Ideal.span
                  (Set.range (ι := σ) (relationOfHasCoeffs.{u_1, u_2, u_3, u_4, u_5} (R := R) (S := S) (P := P) R₀)))) :
            R ⊗[R₀] MvPolynomial ι R₀ →ₐ[R]
              R ⊗[R₀]
                HasQuotient.Quotient (MvPolynomial ι R₀) (B := Ideal (MvPolynomial ι R₀))
                  (Ideal.span
                    (Set.range (ι := σ) (relationOfHasCoeffs.{u_1, u_2, u_3, u_4, u_5} (R := R) (S := S) (P := P) R₀))))
          ((↑(AlgEquiv.symm (MvPolynomial.algebraTensorAlgEquiv R₀ R)) :
              MvPolynomial ι R →ₐ[R] R ⊗[R₀] MvPolynomial ι R₀) :
            MvPolynomial ι R →ₐ[R] R ⊗[R₀] MvPolynomial ι R₀) :
        MvPolynomial ι R →ₐ[R]
          R ⊗[R₀]
            HasQuotient.Quotient (MvPolynomial ι R₀) (B := Ideal (MvPolynomial ι R₀))
              (Ideal.span
                (Set.range (ι := σ) (relationOfHasCoeffs.{u_1, u_2, u_3, u_4, u_5} (R := R) (S := S) (P := P) R₀))))

Tactic:
rw [← P.span_range_relation_eq_ker, Ideal.span_le]

Premises used: Ideal.span_le, Algebra.Presentation.span_range_relation_eq_ker