Theorem about projections and direct sums

Question

I'm in advanced linear algebra and our professor presented this theorem about projections and direct sums:

-If {P₁,...,P_r} is a complete set of orthogonal projections then V=V₁⊕...⊕V_r where V_i=im(P_i).

-Conversely, if V=V₁⊕...⊕V_r, then the map P_i(v₁+...+v_r)=v_i is a projection and {P₁,...,P_r} is a complete set of orthogonal projections.

He provided a proof of the first part that I struggled to follow. Now I am trying to prove the "conversely" part. I think that the steps would be to show that the map is indeed a linear map and projection. Then show that the set is orthogonal and complete. But I'm not sure where to even start with those.