isIntegral_discr_mul_of_mem_traceDual

theorem isIntegral_discr_mul_of_mem_traceDual : ∀ {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [inst : CommRing A] [inst_1 : Field K] [inst_2 : CommRing B] [inst_3 : Field L] [inst_4 : Algebra A K] [inst_5 : Algebra B L] [inst_6 : Algebra A B] [inst_7 : Algebra K L] [inst_8 : Algebra A L] [inst_9 : IsScalarTower A K L] [inst_10 : IsScalarTower A B L] [inst_11 : IsFractionRing A K] [inst_12 : IsIntegralClosure B A L] [inst_13 : FiniteDimensional K L] [inst_14 : IsSeparable K L] [inst_15 : IsIntegrallyClosed A] (I : Submodule B L) {ι : Type u_4} [inst_16 : DecidableEq ι] [inst_17 : Fintype ι] {b : Basis ι K L}, (∀ (i : ι), IsIntegral A (b i)) → ∀ {a x : L}, a ∈ I → x ∈ Submodule.traceDual A K I → IsIntegral A (Algebra.discr K ⇑b • a * x)

This theorem states that if `b` is an `A`-integral basis of a field `L` with discriminant `b`, then the product of `d` times `a` and `x` is an integral element over `A` for all `a` within the submodule `I` of `L` and `x` within the dual space of the `trace` of `I`. Here, `A` and `K` are commutative rings with `K` being the field of fractions of `A`, `B` is an integral closure of `A` in `L`, and `L` is a field which is a finite dimensional `K`-vector space and is separable over `K`. Moreover, `A` is assumed to be integrally closed, and `d` is the discriminant of the basis `b`. The variables `a` and `x` are elements of `L`, and `ι` is a type with decidable equality and finite size used for indexing the basis.

More concisely: If `b` is an integral basis of the separable, finite dimensional field extension `L` over `K`, with integrally closed base ring `A`, then for all `a` in the submodule generated by `b` and `x` in the dual space of `Tr(I)`, their product `da` is an integral element over `A`.