Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u
B : Type u_5
inst✝⁴¹ : CommRing A
inst✝⁴⁰ : Field K
inst✝³⁹ : CommRing B
inst✝³⁸ : Field L
inst✝³⁷ : Algebra A K
inst✝³⁶ : Algebra B L
inst✝³⁵ : Algebra A B
inst✝³⁴ : Algebra K L
inst✝³³ : Algebra A L
inst✝³² : IsScalarTower A K L
inst✝³¹ : IsScalarTower A B L
inst✝³⁰ : IsDomain A
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : FiniteDimensional K L
inst✝²⁷ : Algebra.IsSeparable K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : IsFractionRing B L
inst✝²⁴ : IsIntegrallyClosed A
inst✝²³ : IsDedekindDomain B
C : Type u_3
M : Type u_4
inst✝²² : CommRing C
inst✝²¹ : IsDedekindDomain C
inst✝²⁰ : Field M
inst✝¹⁹ : Algebra C M
inst✝¹⁸ : IsFractionRing C M
inst✝¹⁷ : Algebra A C
inst✝¹⁶ : Algebra B C
inst✝¹⁵ : Algebra A M
inst✝¹⁴ : Algebra B M
inst✝¹³ : Algebra K M
inst✝¹² : Algebra L M
inst✝¹¹ : IsScalarTower A C M
inst✝¹⁰ : IsScalarTower A K M
inst✝⁹ : IsScalarTower B C M
inst✝⁸ : IsScalarTower B L M
inst✝⁷ : IsScalarTower K L M
inst✝⁶ : IsIntegralClosure C A M
inst✝⁵ : FiniteDimensional K M
inst✝⁴ : FiniteDimensional L M
inst✝³ : Algebra.IsSeparable K M
inst✝² : IsIntegralClosure C B M
inst✝¹ : Algebra.IsSeparable L M
inst✝ : IsTorsionFree B C
this : IsLocalization (algebraMapSubmonoid C B⁰) M
h : B⁰ ≤ Submonoid.comap (algebraMap B C) C⁰
h_alg : ∀ {x : L}, (algebraMap L M) x = (IsLocalization.map M (algebraMap B C) h) x
⊢ dual A K 1 = dual B L 1 * (extendedHom M C) (dual A K 1)

Tactic:
refine le_antisymm ?_ ?_

Goal state after:
case refine_1
A : Type u_1
K : Type u_2
L : Type u
B : Type u_5
inst✝⁴¹ : CommRing A
inst✝⁴⁰ : Field K
inst✝³⁹ : CommRing B
inst✝³⁸ : Field L
inst✝³⁷ : Algebra A K
inst✝³⁶ : Algebra B L
inst✝³⁵ : Algebra A B
inst✝³⁴ : Algebra K L
inst✝³³ : Algebra A L
inst✝³² : IsScalarTower A K L
inst✝³¹ : IsScalarTower A B L
inst✝³⁰ : IsDomain A
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : FiniteDimensional K L
inst✝²⁷ : Algebra.IsSeparable K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : IsFractionRing B L
inst✝²⁴ : IsIntegrallyClosed A
inst✝²³ : IsDedekindDomain B
C : Type u_3
M : Type u_4
inst✝²² : CommRing C
inst✝²¹ : IsDedekindDomain C
inst✝²⁰ : Field M
inst✝¹⁹ : Algebra C M
inst✝¹⁸ : IsFractionRing C M
inst✝¹⁷ : Algebra A C
inst✝¹⁶ : Algebra B C
inst✝¹⁵ : Algebra A M
inst✝¹⁴ : Algebra B M
inst✝¹³ : Algebra K M
inst✝¹² : Algebra L M
inst✝¹¹ : IsScalarTower A C M
inst✝¹⁰ : IsScalarTower A K M
inst✝⁹ : IsScalarTower B C M
inst✝⁸ : IsScalarTower B L M
inst✝⁷ : IsScalarTower K L M
inst✝⁶ : IsIntegralClosure C A M
inst✝⁵ : FiniteDimensional K M
inst✝⁴ : FiniteDimensional L M
inst✝³ : Algebra.IsSeparable K M
inst✝² : IsIntegralClosure C B M
inst✝¹ : Algebra.IsSeparable L M
inst✝ : IsTorsionFree B C
this : IsLocalization (algebraMapSubmonoid C B⁰) M
h : B⁰ ≤ Submonoid.comap (algebraMap B C) C⁰
h_alg : ∀ {x : L}, (algebraMap L M) x = (IsLocalization.map M (algebraMap B C) h) x
⊢ dual A K 1 ≤ dual B L 1 * (extendedHom M C) (dual A K 1)

case refine_2
A : Type u_1
K : Type u_2
L : Type u
B : Type u_5
inst✝⁴¹ : CommRing A
inst✝⁴⁰ : Field K
inst✝³⁹ : CommRing B
inst✝³⁸ : Field L
inst✝³⁷ : Algebra A K
inst✝³⁶ : Algebra B L
inst✝³⁵ : Algebra A B
inst✝³⁴ : Algebra K L
inst✝³³ : Algebra A L
inst✝³² : IsScalarTower A K L
inst✝³¹ : IsScalarTower A B L
inst✝³⁰ : IsDomain A
inst✝²⁹ : IsFractionRing A K
inst✝²⁸ : FiniteDimensional K L
inst✝²⁷ : Algebra.IsSeparable K L
inst✝²⁶ : IsIntegralClosure B A L
inst✝²⁵ : IsFractionRing B L
inst✝²⁴ : IsIntegrallyClosed A
inst✝²³ : IsDedekindDomain B
C : Type u_3
M : Type u_4
inst✝²² : CommRing C
inst✝²¹ : IsDedekindDomain C
inst✝²⁰ : Field M
inst✝¹⁹ : Algebra C M
inst✝¹⁸ : IsFractionRing C M
inst✝¹⁷ : Algebra A C
inst✝¹⁶ : Algebra B C
inst✝¹⁵ : Algebra A M
inst✝¹⁴ : Algebra B M
inst✝¹³ : Algebra K M
inst✝¹² : Algebra L M
inst✝¹¹ : IsScalarTower A C M
inst✝¹⁰ : IsScalarTower A K M
inst✝⁹ : IsScalarTower B C M
inst✝⁸ : IsScalarTower B L M
inst✝⁷ : IsScalarTower K L M
inst✝⁶ : IsIntegralClosure C A M
inst✝⁵ : FiniteDimensional K M
inst✝⁴ : FiniteDimensional L M
inst✝³ : Algebra.IsSeparable K M
inst✝² : IsIntegralClosure C B M
inst✝¹ : Algebra.IsSeparable L M
inst✝ : IsTorsionFree B C
this : IsLocalization (algebraMapSubmonoid C B⁰) M
h : B⁰ ≤ Submonoid.comap (algebraMap B C) C⁰
h_alg : ∀ {x : L}, (algebraMap L M) x = (IsLocalization.map M (algebraMap B C) h) x
⊢ dual B L 1 * (extendedHom M C) (dual A K 1) ≤ dual A K 1