Theorem: ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring
File: Mathlib/RingTheory/ClassGroup.lean

Proof steps:
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : DecidableEq L
inst✝⁷ : Algebra R K
inst✝⁶ : IsFractionRing R K
inst✝⁵ : Algebra K L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J : ↥(Ideal R)⁰
⊢ mk0 I = mk0 J ↔ ∃ x, ∃ (_ : x ≠ 0), spanSingleton R⁰ x * ↑↑I = ↑↑J
state_after: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : DecidableEq L
inst✝⁷ : Algebra R K
inst✝⁶ : IsFractionRing R K
inst✝⁵ : Algebra K L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J : ↥(Ideal R)⁰
⊢ (equiv K) (mk0 I) = (equiv K) (mk0 J) ↔ ∃ x, ∃ (_ : x ≠ 0), spanSingleton R⁰ x * ↑↑I = ↑↑J
tactic: refine (ClassGroup.equiv K).injective.eq_iff.symm.trans ?_
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : DecidableEq L
inst✝⁷ : Algebra R K
inst✝⁶ : IsFractionRing R K
inst✝⁵ : Algebra K L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J : ↥(Ideal R)⁰
⊢ (equiv K) (mk0 I) = (equiv K) (mk0 J) ↔ ∃ x, ∃ (_ : x ≠ 0), spanSingleton R⁰ x * ↑↑I = ↑↑J
state_after: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : DecidableEq L
inst✝⁷ : Algebra R K
inst✝⁶ : IsFractionRing R K
inst✝⁵ : Algebra K L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J : ↥(Ideal R)⁰
⊢ (∃ z, (∃ x, spanSingleton R⁰ x = ↑z) ∧ ↑↑I * ↑z = ↑↑J) ↔ ∃ x, x ≠ 0 ∧ spanSingleton R⁰ x * ↑↑I = ↑↑J
tactic: simp only [ClassGroup.equiv_mk0, QuotientGroup.mk'_eq_mk', mem_principal_ideals_iff,
    Units.ext_iff, Units.val_mul, FractionalIdeal.coe_mk0, exists_prop]
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : DecidableEq L
inst✝⁷ : Algebra R K
inst✝⁶ : IsFractionRing R K
inst✝⁵ : Algebra K L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J : ↥(Ideal R)⁰
⊢ (∃ z, (∃ x, spanSingleton R⁰ x = ↑z) ∧ ↑↑I * ↑z = ↑↑J) ↔ ∃ x, x ≠ 0 ∧ spanSingleton R⁰ x * ↑↑I = ↑↑J
state_after: case mp
R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : DecidableEq L
inst✝⁷ : Algebra R K
inst✝⁶ : IsFractionRing R K
inst✝⁵ : Algebra K L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J : ↥(Ideal R)⁰
⊢ (∃ z, (∃ x, spanSingleton R⁰ x = ↑z) ∧ ↑↑I * ↑z = ↑↑J) → ∃ x, x ≠ 0 ∧ spanSingleton R⁰ x * ↑↑I = ↑↑J

case mpr
R : Type u_1
K : Type u_2
L : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : DecidableEq L
inst✝⁷ : Algebra R K
inst✝⁶ : IsFractionRing R K
inst✝⁵ : Algebra K L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
I J : ↥(Ideal R)⁰
⊢ (∃ x, x ≠ 0 ∧ spanSingleton R⁰ x * ↑↑I = ↑↑J) → ∃ z, (∃ x, spanSingleton R⁰ x = ↑z) ∧ ↑↑I * ↑z = ↑↑J
tactic: constructor