Theorem: coeSubmodule_differentIdeal_fractionRing
File: Mathlib/RingTheory/DedekindDomain/Different.lean

tactic: have : IsIntegralClosure B A (FractionRing B) :=
  IsIntegralClosure.of_isIntegrallyClosed _ _ _
state_before: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
⊢ coeSubmodule (FractionRing B) (differentIdeal A B) = 1 / traceDual A (FractionRing A) 1
state_after: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsIntegralClosure B A (FractionRing B)
⊢ coeSubmodule (FractionRing B) (differentIdeal A B) = 1 / traceDual A (FractionRing A) 1

tactic: rw [coeSubmodule, differentIdeal, Submodule.map_comap_eq, inf_eq_right]
state_before: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsIntegralClosure B A (FractionRing B)
⊢ coeSubmodule (FractionRing B) (differentIdeal A B) = 1 / traceDual A (FractionRing A) 1
state_after: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsIntegralClosure B A (FractionRing B)
⊢ 1 / traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))

tactic: have := FractionalIdeal.dual_inv_le (A := A) (K := FractionRing A)
  (1 : FractionalIdeal B⁰ (FractionRing B))
state_before: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsIntegralClosure B A (FractionRing B)
⊢ 1 / traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))
state_after: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝ : IsIntegralClosure B A (FractionRing B)
this : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
⊢ 1 / traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))

tactic: have : _ ≤ ((1 : FractionalIdeal B⁰ (FractionRing B)) : Submodule B (FractionRing B)) := this
state_before: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝ : IsIntegralClosure B A (FractionRing B)
this : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
⊢ 1 / traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))
state_after: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : (fun a => ↑a) (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ ↑1
⊢ 1 / traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))

tactic: simp only [← one_div, FractionalIdeal.val_eq_coe] at this
state_before: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : (fun a => ↑a) (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ ↑1
⊢ 1 / traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))
state_after: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑(1 / FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1
⊢ 1 / traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))

tactic: rw [FractionalIdeal.coe_div (FractionalIdeal.dual_ne_zero _ _ _),
  FractionalIdeal.coe_dual] at this
state_before: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑(1 / FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1
⊢ 1 / traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))
state_after: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑1 / traceDual A (FractionRing A) ↑1 ≤ ↑1
⊢ 1 / traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))

case hI
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑1 / ↑(FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1
⊢ 1 ≠ 0

A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑(1 / FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1
⊢ 1 ≠ 0

tactic: simpa only [FractionalIdeal.coe_one] using this
state_before: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑1 / traceDual A (FractionRing A) ↑1 ≤ ↑1
⊢ 1 / traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))
state_after: no goals

tactic: exact one_ne_zero
state_before: case hI
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑1 / ↑(FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1
⊢ 1 ≠ 0
state_after: no goals

tactic: exact one_ne_zero
state_before: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²³ : CommRing A
inst✝²² : Field K
inst✝²¹ : CommRing B
inst✝²⁰ : Field L
inst✝¹⁹ : Algebra A K
inst✝¹⁸ : Algebra B L
inst✝¹⁷ : Algebra A B
inst✝¹⁶ : Algebra K L
inst✝¹⁵ : Algebra A L
inst✝¹⁴ : IsScalarTower A K L
inst✝¹³ : IsScalarTower A B L
inst✝¹² : IsDomain A
inst✝¹¹ : IsDomain B
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑(1 / FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1
⊢ 1 ≠ 0
state_after: no goals