7. Let the edge length of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ be 1, and the endpoints of the line segment $M N$ be $M$ on the ray $A A_{1}$, and point $N$ on the ray $B C$, and $M N$ intersects the edge $C_{1} D_{1}$ at point $L$. Then the minimum value of $M N$ is $\qquad$
7.3.

Let $A M=x, B N=y$. Then
$$
M N=\sqrt{x^{2}+y^{2}+1} \text {. }
$$

Draw $L L_{1} \perp D C$ at point $L_{1}$. Thus,
$$
\begin{array}{l}
\frac{1}{x-1}=\frac{A A_{1}}{M A_{1}}=\frac{L M}{L N}=\frac{N C}{C B}=\frac{y-1}{1} \\
\Rightarrow x y=x+y \\
\Rightarrow x^{2} y^{2} \geqslant 4 x y \\
\Rightarrow x y \geqslant 4 .
\end{array}
$$

Therefore, $M N=\sqrt{x^{2}+y^{2}+1} \geqslant \sqrt{2 x y+1} \geqslant 3$,
with equality holding if and only if $x=y=2$.