-- Module: Mathlib.RingTheory.Localization.AtPrime.Basic
Goal:
R_1 : Type u_1
inst : CommSemiring R_1
S_1 : Type u_2
inst_1 : CommSemiring S_1
inst_2 : Algebra R_1 S_1
P_1 : Type u_3
inst_3 : CommSemiring P_1
p_1 : Ideal R_1
inst_4 : p_1.IsPrime
Rₚ_1 : Type u_4
inst_5 : CommSemiring Rₚ_1
inst_6 : Algebra R_1 Rₚ_1
inst_7 : IsLocalization.AtPrime Rₚ_1 p_1
inst_8 : IsLocalRing Rₚ_1
Sₚ_1 : Type u_5
inst_9 : CommSemiring Sₚ_1
inst_10 : Algebra S_1 Sₚ_1
inst_11 : IsLocalization (algebraMapSubmonoid S_1 p_1.primeCompl) Sₚ_1
inst_12 : Algebra Rₚ_1 Sₚ_1
P : Ideal S_1
S : Type u_6
R : Type u_7
inst_13 : CommRing R
p : Ideal R
inst_14 : p.IsMaximal
Rₚ : Type u_8
inst_15 : CommRing Rₚ
inst_16 : Algebra R Rₚ
inst_17 : IsLocalization.AtPrime Rₚ p
inst_18 : IsLocalRing Rₚ
Sₚ : Type u_9
inst_19 : CommRing S
inst_20 : Algebra R S
inst_21 : CommRing Sₚ
inst_22 : Algebra S Sₚ
inst_23 : Algebra R Sₚ
inst_24 : Algebra Rₚ Sₚ
inst_25 : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst_26 : IsScalarTower R S Sₚ
inst_27 : IsScalarTower R Rₚ Sₚ
h :
  Eq.{u_9 + 1} (α := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))
    (Ideal.map (R := S) (F := S →+* Sₚ) (algebraMap S Sₚ) (Ideal.map (F := R →+* S) (algebraMap R S) p))
refine_1 :
  Eq (α := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)
    (RingHom.ker (F :=
      RingHom.{u_6, u_9} S
        (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
      (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
          (algebraMap Sₚ
            (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
          (algebraMap S Sₚ) :
        RingHom.{u_6, u_9} S
          (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))))
refine_1₁ :
  Eq (α := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)
    (RingHom.ker (F :=
      RingHom.{u_6, u_9} S
        (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
      (RingHom.comp (Ideal.Quotient.mk (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))
          (algebraMap S Sₚ) :
        RingHom.{u_6, u_9} S
          (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))))
refine_1₂ :
  Eq (α := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)
    (comap (F := S →+* Sₚ) (algebraMap S Sₚ : S →+* Sₚ)
      (RingHom.ker (F :=
        Sₚ →+*
          HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))
        (Ideal.Quotient.mk (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))))
refine_1₃ :
  Eq (α := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)
    (comap (F := S →+* Sₚ) (algebraMap S Sₚ : S →+* Sₚ)
      (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))
refine_1₄ :
  Eq (α := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)
    (comap (F := S →+* Sₚ) (algebraMap S Sₚ : S →+* Sₚ)
      (Ideal.map (R := S) (F := S →+* Sₚ) (algebraMap S Sₚ) (Ideal.map (F := R →+* S) (algebraMap R S) p)))
refine_1₅ :
  Eq (α := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)
    (comap (F := S →+* Sₚ) (algebraMap S Sₚ : S →+* Sₚ)
      (Ideal.map (F := R →+* Sₚ)
        (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S Sₚ) (algebraMap R S)) p))
refine_1₆ :
  Eq (α := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)
    (comap (F := S →+* Sₚ) (algebraMap S Sₚ : S →+* Sₚ) (Ideal.map (F := R →+* Sₚ) (algebraMap R Sₚ) p))
refine_1₇ :
  Eq (α := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p : Ideal S)
    (under.{u_6, u_9} S (B := Sₚ) (Ideal.map (F := R →+* Sₚ) (algebraMap R Sₚ) p))
refine_1₈ refine_1₉ :
  Eq.{u_6 + 1} (α := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)
    (Ideal.map (F := R →+* S) (algebraMap R S) p)
⊢ Eq (α := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) p)
    (RingHom.ker (F :=
      RingHom.{u_6, u_9} S
        (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))))
      (algebraMap S
          (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ))) :
        RingHom.{u_6, u_9} S
          (HasQuotient.Quotient Sₚ (B := Ideal Sₚ) (Ideal.map (F := Rₚ →+* Sₚ) (algebraMap Rₚ Sₚ) (maximalIdeal Rₚ)))))

Tactic:
rw [IsScalarTower.algebraMap_eq S Sₚ, Ideal.Quotient.algebraMap_eq, ← RingHom.comap_ker,
      Ideal.mk_ker, h, Ideal.map_map, ← IsScalarTower.algebraMap_eq, ← under_def, under_map_eq_map]

Premises used: Ideal.Quotient.algebraMap_eq, RingHom.comap_ker, IsLocalization.AtPrime.under_map_eq_map, Ideal.under_def, Ideal.map_map, IsScalarTower.algebraMap_eq, Ideal.mk_ker