Question about isomorphic mapping on direct sums?

You're given:
- [itex]V = \bigoplus_{a\in A} V_a[/itex], where [itex]A[/itex] is some possibly infinite index set.
- For each [itex]a\in A[/itex], the inclusion map [itex]\iota_a: V_a \to V[/itex] given by [itex]\iota_a(x_a) = (x_a, (0_b)_{b\in A\setminus\{a\}})[/itex].
- For each [itex]a\in A[/itex], the projection map [itex]\pi_a: V \to V_a[/itex] given by [itex]\pi_a(x) = (x_a)[/itex].

You've noticed that, if [itex]|A|<\infty[/itex], then [itex]\sum_{a\in A} \iota_a\circ\pi_a = id_V: V\to V[/itex].

So you're asking whether the condition that [itex]|A|<\infty[/itex] can be dropped to draw the same conclusion? The answer is essentially
1) Yes.
2) One has to be careful about the definition of [itex]\sum_{a\in A}[/itex] when [itex]A[/itex] is infinite.
3) There's very little content here, as point (2) is essentially built into the definition of a direct sum.