Theorem: ClassGroup.induction
File: Mathlib/RingTheory/ClassGroup.lean

Proof steps:
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : DecidableEq L
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : Algebra K L
inst✝³ : FiniteDimensional K L
inst✝² : Algebra R L
inst✝¹ : IsScalarTower R K L
inst✝ : IsDomain R
P : ClassGroup R → Prop
h : ∀ (I : (FractionalIdeal R⁰ K)ˣ), P (mk I)
x : ClassGroup R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
⊢ P ↑I
state_after: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : DecidableEq L
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : Algebra K L
inst✝³ : FiniteDimensional K L
inst✝² : Algebra R L
inst✝¹ : IsScalarTower R K L
inst✝ : IsDomain R
P : ClassGroup R → Prop
h : ∀ (I : (FractionalIdeal R⁰ K)ˣ), P (mk I)
x : ClassGroup R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
this :
  I =
    (Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv)
      ((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv) I)
⊢ P ↑I
tactic: have : I = (Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv)
      (Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv I) := by
      simp [← Units.eq_iff]
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : DecidableEq L
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : Algebra K L
inst✝³ : FiniteDimensional K L
inst✝² : Algebra R L
inst✝¹ : IsScalarTower R K L
inst✝ : IsDomain R
P : ClassGroup R → Prop
h : ∀ (I : (FractionalIdeal R⁰ K)ˣ), P (mk I)
x : ClassGroup R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
this :
  I =
    (Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv)
      ((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv) I)
⊢ P ↑I
state_after: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : DecidableEq L
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : Algebra K L
inst✝³ : FiniteDimensional K L
inst✝² : Algebra R L
inst✝¹ : IsScalarTower R K L
inst✝ : IsDomain R
P : ClassGroup R → Prop
h : ∀ (I : (FractionalIdeal R⁰ K)ˣ), P (mk I)
x : ClassGroup R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
this :
  I =
    (Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv)
      ((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv) I)
⊢ P
    ↑((Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv)
        ((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv) I))
tactic: rw [congr_arg (QuotientGroup.mk (s := (toPrincipalIdeal R (FractionRing R)).range)) this]
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : DecidableEq L
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : Algebra K L
inst✝³ : FiniteDimensional K L
inst✝² : Algebra R L
inst✝¹ : IsScalarTower R K L
inst✝ : IsDomain R
P : ClassGroup R → Prop
h : ∀ (I : (FractionalIdeal R⁰ K)ˣ), P (mk I)
x : ClassGroup R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
this :
  I =
    (Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv)
      ((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv) I)
⊢ P
    ↑((Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv)
        ((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv) I))
state_after: no goals
tactic: exact h _