Theorem: Ideal.natAbs_det_equiv
File: Mathlib/RingTheory/Ideal/Norm.lean

Proof steps:
state_before: S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case pos
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : I = ⊥
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I

case neg
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: by_cases hI : I = ⊥
state_before: case neg
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: let ι := Module.Free.ChooseBasisIndex ℤ S
state_before: case neg
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: let b := Module.Free.chooseBasis ℤ S
state_before: case neg
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inl
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : IsEmpty ι
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I

case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: cases isEmpty_or_nonempty ι
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: letI := Ideal.fintypeQuotientOfFreeOfNeBot I hI
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: letI := Classical.decEq ι
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: let a := I.smithCoeffs b hI
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: let b' := I.ringBasis b hI
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: let ab := I.selfBasis b hI
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: have ab_eq := I.selfBasis_def b hI
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: let e' : S ≃ₗ[ℤ] I := b'.equiv ab (Equiv.refl _)
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: let f : S →ₗ[ℤ] S := (I.subtype.restrictScalars ℤ).comp (e' : S →ₗ[ℤ] I)
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
tactic: let f_apply : ∀ x, f x = b'.equiv ab (Equiv.refl _) x := fun x => rfl
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
⊢ (LinearMap.det f).natAbs = absNorm I
tactic: suffices (LinearMap.det f).natAbs = Ideal.absNorm I by
    calc
      _ = (LinearMap.det ((Submodule.subtype I).restrictScalars ℤ ∘ₗ
            (AddEquiv.toIntLinearEquiv e : S ≃ₗ[ℤ] I))).natAbs := rfl
      _ = (LinearMap.det ((Submodule.subtype I).restrictScalars ℤ ∘ₗ _)).natAbs :=
            Int.natAbs_eq_iff_associated.mpr (LinearMap.associated_det_comp_equiv _ _ _)
      _ = absNorm I := this
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
⊢ (LinearMap.det f).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
ha : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), f (b' i) = a i • b' i
⊢ (LinearMap.det f).natAbs = absNorm I
tactic: have ha : ∀ i, f (b' i) = a i • b' i := by
    intro i; rw [f_apply, b'.equiv_apply, Equiv.refl_apply, ab_eq]
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
ha : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), f (b' i) = a i • b' i
⊢ (LinearMap.det f).natAbs = absNorm I
state_after: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝¹ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this✝ : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
ha : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), f (b' i) = a i • b' i
this : DecidableEq ι := Classical.decEq ι
⊢ (LinearMap.det f).natAbs = absNorm I
tactic: letI := Classical.decEq ι
state_before: case neg.inr
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝¹ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this✝ : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
ha : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), f (b' i) = a i • b' i
this : DecidableEq ι := Classical.decEq ι
⊢ (LinearMap.det f).natAbs = absNorm I
state_after: case neg.inr.calc_1
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝¹ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this✝ : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
ha : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), f (b' i) = a i • b' i
this : DecidableEq ι := Classical.decEq ι
⊢ ((LinearMap.toMatrix b' b') f).det.natAbs = (Matrix.diagonal a).det.natAbs

case neg.inr.calc_2
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝¹ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this✝ : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
ha : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), f (b' i) = a i • b' i
this : DecidableEq ι := Classical.decEq ι
⊢ ∏ i : Module.Free.ChooseBasisIndex ℤ S, (a i).natAbs = Nat.card (S ⧸ I)
tactic: calc
    Int.natAbs (LinearMap.det f) = Int.natAbs (LinearMap.toMatrix b' b' f).det := by
      rw [LinearMap.det_toMatrix]
    _ = Int.natAbs (Matrix.diagonal a).det := ?_
    _ = Int.natAbs (∏ i, a i) := by rw [Matrix.det_diagonal]
    _ = ∏ i, Int.natAbs (a i) := map_prod Int.natAbsHom a Finset.univ
    _ = Nat.card (S ⧸ I) := ?_
    _ = absNorm I := (Submodule.cardQuot_apply _).symm
state_before: case neg.inr.calc_2
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝¹ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this✝ : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
ha : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), f (b' i) = a i • b' i
this : DecidableEq ι := Classical.decEq ι
⊢ ∏ i : Module.Free.ChooseBasisIndex ℤ S, (a i).natAbs = Nat.card (S ⧸ I)
state_after: case neg.inr.calc_2
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝² : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this✝¹ : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
ha : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), f (b' i) = a i • b' i
this✝ : DecidableEq ι := Classical.decEq ι
this : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), NeZero (a i).natAbs
⊢ ∏ i : Module.Free.ChooseBasisIndex ℤ S, (a i).natAbs = Nat.card (S ⧸ I)
tactic: haveI : ∀ i, NeZero (a i).natAbs := fun i =>
    ⟨Int.natAbs_ne_zero.mpr (Ideal.smithCoeffs_ne_zero b I hI i)⟩
state_before: case neg.inr.calc_2
S : Type u_1
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S
h✝ : Nonempty ι
this✝² : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI
this✝¹ : DecidableEq ι := Classical.decEq ι
a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI
b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI
ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI
ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))
f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'
f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl
ha : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), f (b' i) = a i • b' i
this✝ : DecidableEq ι := Classical.decEq ι
this : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), NeZero (a i).natAbs
⊢ ∏ i : Module.Free.ChooseBasisIndex ℤ S, (a i).natAbs = Nat.card (S ⧸ I)
state_after: no goals
tactic: simp_rw [Nat.card_congr (Ideal.quotientEquivPiZMod I b hI).toEquiv, Nat.card_pi,
    Nat.card_zmod]