Proofs of abelian group axioms for elliptic curve over finite fields + tangent-chord group law

Question

I'm looking for the proofs of the abelian group axioms for elliptic curve over finite fields (e.g. integers mod p) with the tangent-chord group law (i.e. the "standard" group law for elliptic curves) as the group operation:

1. Closure
2. Associativity
3. Identity element
4. Inverse element
5. Commutativity

References to papers or books are fine, as well as multiple proofs per axiom. I have done some research and have found some myself, but I've come up with nothing especially regarding closure.

I'm a physicist by training, so it might entirely be possible that I overlooked some well-known connections.

Answer 1

One very popular textbook in the field is https://www.springer.com/gp/book/9780387094939 by Silverman. Section III.2 contains proofs of closure, identity, commutativity and inverse in Proposition 2.2. Associativity is only sortof proved there, but there are references to different approaches to the proof there.