feat(ring_theory/integral_closure): Rings lying within an integral extension are integral extensions (#3781)
feat(ring_theory/integral_closure): Rings lying within an integral extension are integral extensions (#3781)

Proofs that if an algebra tower is integral then so are the two extensions making up the tower. I needed these for another proof I'm working on, but it seemed reasonable to create a smaller PR now since they are basically self contained.

Co-authored-by: Devon Tuma <e0b2a7dc618b27b0ecce10eb436e37b1588f54f4@gmail.com>

leanprover-community/mathlib,leanprover-community/mathlib