If $T: V \to V$ is a linear operator such that $T^5 = T^4$, then $\ker T^4 \cap Im\ T^4 $ is a singleton. $\renewcommand{\Im}{\operatorname{Im}}$I have to prove or disprove

If $T: V \to V$ is a linear operator such that $T^5 = T^4$, then  $\ker
 T^4 \cap \Im T^4 $ is  a singleton. Here $\ker$ stands for the kernel
  and $\Im$ stands for the image space of $V$ respectively. Can we also show that $V$ is direct sum of $\ker T^4$ and $\Im T^4$.

My approach Let $x \in \ker
T^4 \cap \Im T^4$. Then $x \in \ker
T^4 $ and  $\Im T^4$. Now $x \in \ker
T^4 $ implies that $T^4(x) = 0$ and $x \in \Im T^4$ implies that there must exist  $w \in V$ such that $T^4(w) = x$. From here I am not able to proceed. I need help.
Thanks
 A: An answer inspired in no small part by the comments of our colleague flan:
We observe that
$(T^4)^2 = T^8 = T^5T^3 = T^4T^3 = T^5T^2$
$= T^4T^2 = T^5T = T^4T = T^5 = T^4;   \tag 1$
that is, $T^4$ is idempotent; for idempotent operators $P$ we have
$P^2 = P,   \tag 2$
and this implies 
$\ker P \cap \text{Im} P = \{0\}, \tag 3$
for if
$y \in \ker P \cap \text{Im} P,   \tag 4$
then
$\exists z, \; y = Pz \tag 5$
and
$Py = 0; \tag 6$
then
$y = Pz = P^2z = PPz = Py = 0, \tag 7$
and we have (3); with
$P = T^4, \tag 8$
this shows that
$\ker T^4 \cap \text{Im} T^4 = \{0\}, \tag 9$
a singleton.
We also have, for any vector $y$ 
$y = y - Py + Py = (I - P)y + Py, \tag{10}$
and with $P$ idempotent
$P(I - P)y = (P - P^2)y = 0, \tag{11}$
that is
$(I - P)y \in \ker P,  \tag{12}$
whence via (10),
$y \in \ker P + \text{Im} P, \tag{13}$
and this in concert with (3) shows that
$V = \ker P \oplus \text{Im} P; \tag{14}$
since $T^4$ is idempotent, it follows that
$V = \ker T^4 \oplus \text{Im} T^4, \tag{15}$
as per request.  $OE\Delta$.
A: By reduction to absurd.
We suposse that exists $0 \neq a \in Ker T^{4} \cap Im T^{4}$. Then we considerer $w \in (T^{4})^{-1}(a)$ and we have that:
$a=T^{4}(w)=T^{5}(w)=T(T^{4}(w))=T(a)$
So we have that $0=T^{4}(a)=a$ but $a \neq 0$. ABSURD!!!!
So $Ker T^{4} \cap Im T^{4} = \{0\}$ and is a singleton.
