Proof that A+A projects onto R(A) and AA+ projects onto N(A)

Question

I have been able to understand the proof that shows both are projections by proving P^2 = P for both of them. I don't understand how these projections project onto R(A) and C(A) though.

Proof that they are projections by showing P^2 = P for both of them

I am aware that R(A) spans {v1 ... vr} (non-zero eigenmatrix for ATA) and C(A) spans {vr+1 ... vn} (zero eigenmatrix for ATA); do they play a part in answering this question?

Thank you for any help.