No dots in the cross-reference to an item from enumerate

Question

I defined the labels of items from enumerate as following:

    1.\usepackage{enumerate}
    2.\renewcommand{\theenumi}{\arabic{enumi}}
    3.\renewcommand{\labelenumi}{\theenumi} 
    4.\renewcommand{\theenumii}{\arabic{enumii}}
    5.\renewcommand{\labelenumii}{\theenumi.~\theenumii}
    6.\renewcommand{\theenumiii}{\arabic{enumiii}}
    7.\renewcommand{\labelenumiii}{\theenumi.~\theenumii.~\theenumiii}

I can create the enumeration of items on three levels with dots:

1
1.1
1.1.1

I reference the item with the command \ref{item_label}

\item \label{item_label} item_text
In \ref{item_label} you can find ...

The numbers in enumeration have dots, but in the reference the dots are missing, although I defined them in lines 5 and 7.

1.1.1 item_text
In 111 you can find ...  

How can I have dots in the cross-references? I want to use only enumerate, and no other package.

Answer 1

\documentclass{article}

\usepackage{enumerate}
   \renewcommand{\theenumi}{\arabic{enumi}}
   \renewcommand{\labelenumi}{\theenumi} 
   \renewcommand{\theenumii}{\arabic{enumii}}
   \renewcommand{\labelenumii}{\theenumi.~\theenumii}
   \renewcommand{\theenumiii}{\arabic{enumiii}}
   \renewcommand{\labelenumiii}{\theenumi.~\theenumii.~\theenumiii}

\makeatletter
\renewcommand\p@enumii{\theenumi.}
\renewcommand\p@enumiii{\theenumi.\theenumi.}
\makeatother

\begin{document}

\begin{enumerate}
\item xxx
\begin{enumerate}
\item xxx
\begin{enumerate}
\item xx \label{x}
\end{enumerate}
\end{enumerate}
\end{enumerate}

see [\ref{x}]

\end{document}