-- Mathlib/RingTheory/Ideal/GoingUp.lean
theorem exists_ideal_over_prime_of_isIntegral_of_isDomain [Algebra.IsIntegral R S] (P : Ideal R)
    [IsPrime P] (hP : RingHom.ker (algebraMap R S) ≤ P) :
    ∃ Q : Ideal S, IsPrime Q ∧ Q.comap (algebraMap R S) = P

Proof state:
case intro
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
Sₚ : Type u_2 := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)
this : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) :=
  IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)
Qₚ : Ideal Sₚ
Qₚ_maximal : Qₚ.IsMaximal
⊢ ∃ Q, Q.IsPrime ∧ comap (algebraMap R S) Q = P

Tactic:
let _ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S

Result:
case intro
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
Sₚ : Type u_2 := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)
this : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) :=
  IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)
Qₚ : Ideal Sₚ
Qₚ_maximal : Qₚ.IsMaximal
x✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
⊢ ∃ Q, Q.IsPrime ∧ comap (algebraMap R S) Q = P

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
Sₚ : Type u_2 := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)
this : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) :=
  IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)
⊢ ∃ Q, Q.IsPrime ∧ comap (algebraMap R S) Q = P

Tactic:
obtain ⟨Qₚ : Ideal Sₚ, Qₚ_maximal⟩ := exists_maximal Sₚ

Result:
case intro
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
Sₚ : Type u_2 := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)
this : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) :=
  IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)
Qₚ : Ideal Sₚ
Qₚ_maximal : Qₚ.IsMaximal
⊢ ∃ Q, Q.IsPrime ∧ comap (algebraMap R S) Q = P

Proof state:
case h.e'_2
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
Sₚ : Type u_2 := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)
this✝ : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) :=
  IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)
Qₚ : Ideal Sₚ
Qₚ_maximal : Qₚ.IsMaximal
x✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : Algebra.IsIntegral Rₚ Sₚ
Qₚ_max : (comap (algebraMap Rₚ Sₚ) Qₚ).IsMaximal
⊢ comap (algebraMap R S) (comap (algebraMap S Sₚ) Qₚ) =
    comap (algebraMap R (Localization.AtPrime P)) (IsLocalRing.maximalIdeal (Localization P.primeCompl))

Tactic:
rw [comap_comap, ← IsLocalRing.eq_maximalIdeal Qₚ_max,
  ← IsLocalization.map_comp (P := S) (Q := Sₚ) (g := algebraMap R S)
  (M := P.primeCompl) (T := Algebra.algebraMapSubmonoid S P.primeCompl) (S := Rₚ)
  (fun p hp => Algebra.mem_algebraMapSubmonoid_of_mem ⟨p, hp⟩) ]

Result:
case h.e'_2
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
Sₚ : Type u_2 := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)
this✝ : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) :=
  IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)
Qₚ : Ideal Sₚ
Qₚ_maximal : Qₚ.IsMaximal
x✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : Algebra.IsIntegral Rₚ Sₚ
Qₚ_max : (comap (algebraMap Rₚ Sₚ) Qₚ).IsMaximal
⊢ comap ((IsLocalization.map Sₚ (algebraMap R S) ⋯).comp (algebraMap R Rₚ)) Qₚ =
    comap (algebraMap R (Localization.AtPrime P)) (comap (algebraMap Rₚ Sₚ) Qₚ)

Proof state:
case intro
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
Sₚ : Type u_2 := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)
this✝ : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) :=
  IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)
Qₚ : Ideal Sₚ
Qₚ_maximal : Qₚ.IsMaximal
x✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : Algebra.IsIntegral Rₚ Sₚ
Qₚ_max : (comap (algebraMap Rₚ Sₚ) Qₚ).IsMaximal
⊢ comap (algebraMap R S) (comap (algebraMap S Sₚ) Qₚ) = P

Tactic:
convert Localization.AtPrime.comap_maximalIdeal (I := P)

Result:
case h.e'_2
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
Sₚ : Type u_2 := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)
this✝ : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) :=
  IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)
Qₚ : Ideal Sₚ
Qₚ_maximal : Qₚ.IsMaximal
x✝ : Algebra Rₚ Sₚ := localizationAlgebra P.primeCompl S
this : Algebra.IsIntegral Rₚ Sₚ
Qₚ_max : (comap (algebraMap Rₚ Sₚ) Qₚ).IsMaximal
⊢ comap (algebraMap R S) (comap (algebraMap S Sₚ) Qₚ) =
    comap (algebraMap R (Localization.AtPrime P)) (IsLocalRing.maximalIdeal (Localization P.primeCompl))

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
⊢ ∃ Q, Q.IsPrime ∧ comap (algebraMap R S) Q = P

Tactic:
let Sₚ := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)

Result:
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDomain S
inst✝¹ : Algebra.IsIntegral R S
P : Ideal R
inst✝ : P.IsPrime
hP : RingHom.ker (algebraMap R S) ≤ P
hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization P.primeCompl
Sₚ : Type u_2 := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)
⊢ ∃ Q, Q.IsPrime ∧ comap (algebraMap R S) Q = P