Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: case refine_1
R : Type u_1
inst✝²⁴ : CommRing R
inst✝²³ : IsDomain R
S : Type u_3
inst✝²² : CommRing S
inst✝²¹ : IsDomain S
inst✝²⁰ : IsIntegrallyClosed R
inst✝¹⁹ : IsIntegrallyClosed S
inst✝¹⁸ : Algebra R S
inst✝¹⁷ : Module.Finite R S
inst✝¹⁶ : IsTorsionFree R S
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹⁵ : CommRing Rₘ
inst✝¹⁴ : Algebra R Rₘ
inst✝¹³ : CommRing Sₘ
inst✝¹² : Algebra S Sₘ
inst✝¹¹ : Algebra Rₘ Sₘ
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : IsScalarTower R Rₘ Sₘ
inst✝⁸ : IsScalarTower R S Sₘ
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁵ : IsIntegrallyClosed Rₘ
inst✝⁴ : IsDomain Rₘ
inst✝³ : IsDomain Sₘ
inst✝² : IsTorsionFree Rₘ Sₘ
inst✝¹ : Module.Finite Rₘ Sₘ
inst✝ : IsIntegrallyClosed Sₘ
K : Type u_1 := FractionRing R
f : Rₘ →+* K := IsLocalization.map K (RingHom.id R) hM
L : Type u_3 := FractionRing S
g : Sₘ →+* L := IsLocalization.map L (RingHom.id S) ⋯
algInst✝² : Algebra Rₘ K := f.toAlgebra
algInst✝¹ : Algebra Sₘ L := g.toAlgebra
algInst✝ : Algebra Rₘ L := ((algebraMap K L).comp f).toAlgebra
scalarTowerInst✝ : IsScalarTower Rₘ K L := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Rₘ L))
this✝¹ : IsScalarTower R Rₘ K
x✝¹ : IsFractionRing Rₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Rₘ K
this✝ : IsScalarTower S Sₘ L
this : IsScalarTower Rₘ Sₘ L
x✝ : IsFractionRing Sₘ L :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization (Algebra.algebraMapSubmonoid S M) Sₘ L
a' : Sₘ
a : S
ha : a ∈ I
s : R
hs : s ∈ ↑M
has : a' * (algebraMap S Sₘ) ↑⟨(algebraMap R S) s, ⋯⟩ = (algebraMap S Sₘ) ↑⟨a, ha⟩
⊢ ∃ a b, (Algebra.intNorm Rₘ Sₘ) a' * (algebraMap R Rₘ) ↑b = (algebraMap R Rₘ) ↑a

Tactic:
refine ⟨⟨Algebra.intNorm R S a, intNorm_mem_spanNorm _ ha⟩,
      ⟨s ^ Module.finrank K L, pow_mem hs _⟩, ?_⟩

Goal state after:
case refine_1
R : Type u_1
inst✝²⁴ : CommRing R
inst✝²³ : IsDomain R
S : Type u_3
inst✝²² : CommRing S
inst✝²¹ : IsDomain S
inst✝²⁰ : IsIntegrallyClosed R
inst✝¹⁹ : IsIntegrallyClosed S
inst✝¹⁸ : Algebra R S
inst✝¹⁷ : Module.Finite R S
inst✝¹⁶ : IsTorsionFree R S
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹⁵ : CommRing Rₘ
inst✝¹⁴ : Algebra R Rₘ
inst✝¹³ : CommRing Sₘ
inst✝¹² : Algebra S Sₘ
inst✝¹¹ : Algebra Rₘ Sₘ
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : IsScalarTower R Rₘ Sₘ
inst✝⁸ : IsScalarTower R S Sₘ
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁵ : IsIntegrallyClosed Rₘ
inst✝⁴ : IsDomain Rₘ
inst✝³ : IsDomain Sₘ
inst✝² : IsTorsionFree Rₘ Sₘ
inst✝¹ : Module.Finite Rₘ Sₘ
inst✝ : IsIntegrallyClosed Sₘ
K : Type u_1 := FractionRing R
f : Rₘ →+* K := IsLocalization.map K (RingHom.id R) hM
L : Type u_3 := FractionRing S
g : Sₘ →+* L := IsLocalization.map L (RingHom.id S) ⋯
algInst✝² : Algebra Rₘ K := f.toAlgebra
algInst✝¹ : Algebra Sₘ L := g.toAlgebra
algInst✝ : Algebra Rₘ L := ((algebraMap K L).comp f).toAlgebra
scalarTowerInst✝ : IsScalarTower Rₘ K L := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Rₘ L))
this✝¹ : IsScalarTower R Rₘ K
x✝¹ : IsFractionRing Rₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Rₘ K
this✝ : IsScalarTower S Sₘ L
this : IsScalarTower Rₘ Sₘ L
x✝ : IsFractionRing Sₘ L :=
  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization (Algebra.algebraMapSubmonoid S M) Sₘ L
a' : Sₘ
a : S
ha : a ∈ I
s : R
hs : s ∈ ↑M
has : a' * (algebraMap S Sₘ) ↑⟨(algebraMap R S) s, ⋯⟩ = (algebraMap S Sₘ) ↑⟨a, ha⟩
⊢ (Algebra.intNorm Rₘ Sₘ) a' * (algebraMap R Rₘ) ↑⟨s ^ finrank K L, ⋯⟩ = (algebraMap R Rₘ) ↑⟨(Algebra.intNorm R S) a, ⋯⟩