Showing module can be decomposed

Question

Let $M$ be an $R$-module and $\pi_1,\dots,\pi_k \in \operatorname{End}_R (M)$ such that the following conditions are satisfied:
1) $\pi_i ^2=\pi_i$;
2) $\sum_{i=1}^{k} \pi_i $ is the identity map;
3) $\pi_i \pi_j = 0$ for $i \neq j$.
Set $M_i = \operatorname{Im}(\pi_i)$.
I need to show that $M \cong M_1 \oplus \dots \oplus M_k$.
I know there is a theorem saying that if $M$ is an $R$-module and $M_1,\dots,M_k$ are submodules of $M$, then $M \cong M_1 \oplus \dots \oplus M_k$ if and only if $M=M_1+\dots+M_k$ and $M_j \cap \sum_{\substack{i=1, \\ i\neq j}}^{k} M_i = \{0_M\}$.
How can show what I want to prove is equivalent to the conditions from the theorem?

Answer 1

Hint:

Relation (2) says that for any $m\in M$, $m=\sum_{i=1}^k\pi_i(m)\in\sum_{i=1}^kM_i$.

There remains to show that $\;Mj\cap\sum\limits_{\smash[b]{\substack{i=1\\i\ne j}}^k}Mi=\{0\}$. Suppose that $$m_j=\sum\limits_{\substack{i=1\\i\ne j}}^k m_i,\qquad (m_j\in M_j, m_i\in M_i) $$ and apply $\pi_j$ to both sides.