This is going to be an exciting area. I spent some time previously playing with Coq. It's very powerful, but even proving the simplest undergraduate maths statements (say, about group theory) can prove very challenging. I believe that part of this is that Coq uses different mathematical foundations than traditional mathematics, which mostly uses set theory (ZFC, although most people don't care about the specifics). S…
Given a group G, there is a type `subgroup G` of subgroups of G, and a subgroup is a structure consisting of a set of elements of G along with some proofs that it has the identity, it's closed under multiplication, and it's closed under taking inverses. Lean has a coercion facility using typeclasses, and there is a typeclass instance that will use the subgroup's set coerced to a type when the subgroup is used in a place that expects a type. This coerced type has a group typeclass instance, so a subgroup "is" a group.
The big complexity in all of this doesn't seem to be ZFC vs type theory, but rather how do you implement a mathematician's use of synecdoche and implicit coercions. Synecdoche is the figure of speech where you refer to a thing by its parts or vice versa -- for example, "let G be a group" and then using G as if it were a set, even though a group consists of a set, a binary operation on the set, a choice of identity element, and a few axioms that must hold. Mathlib uses typeclasses to implement the synecdoche -- a type is a group if there is a typeclass instance that provides the rest of the structure. As I understand it, Coq's Mathematical Components library uses structures and unification hints instead (though I've failed to understand how it works exactly), but I have heard that they can be very fiddly to get right.
I think you'd have to find solutions to these same problems no matter the foundations. At least with types, there's enough information lying around that, for example, group structures can be automatically synthesized by the typeclass resolution system in many common situations.