Numbering sublemmas

I would like to have sublemmas numbered after the corresponding lemmas, which in turn would be numbered after theorems, like this:

Theorem 2.1. 0=0.

Lemma 2.2. 0=1.

The proof of Lemma 2.2 will be given later.

Lemma 2.3. 1=1.

Sublemma 2.2.1. 0=2.

Proof: Obvious.

Sublemma 2.2.2. 0=4.

Proof: Obvious.

Sublemma 2.3.1. 0=0.

Here Sublemma 2.2.1 and 2.2.2 are sublemmas of Lemma 2.2 (even though in the text they appear only after Lemma 2.3), and Sublemma 2.3.1 is a sublemma of Lemma 2.3.

I need this, because I would like to first present, in a series of lemmas, a general outline of the proof of a theorem, and then prove the lemmas via sublemmas, each of the daughter sublemmas clearly corresponding to its parent lemma.

Is this possible to do?

The solutions offered at Lemmas and sub sections do make "the sublemma counter subsidiary to the lemma counter". However, there the sublemmas of a lemma (say Lemma 5) must follow Lemma 5, with no other lemmas stated between Lemma 5 and its sublemmas.

I have found the following elementary, yet cumbersome solution. I'd still appreciate a better one.

\documentclass[]{article}

\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}

\newcounter{sublOne}
\newenvironment{sublOne}[]{\refstepcounter{sublOne}\par\medskip
\noindent \textbf{Sublemma~\ref{lem:1}.\thesublOne. #1} \rmfamily}{\medskip}

\newcounter{sublTwo}
\newenvironment{sublTwo}[]{\refstepcounter{sublTwo}\par\medskip
\noindent \textbf{Sublemma~\ref{lem:2}.\thesublTwo. #1} \rmfamily}{\medskip}

\newcommand{\refs}{\ref{#1}.\ref{#2}}

\begin{document}

\section{First section}
\begin{thm} $0=0$. \end{thm}

------------------------------

\section{Second section}

\begin{thm} $1=1$. \end{thm}
\begin{lem}\label{lem:1} $2=2$ \end{lem}
\begin{lem}\label{lem:2} $3=3$ \end{lem}

\begin{sublOne}\label{subl:2.2.1}
$2.1=2.1$. \end{sublOne}

\begin{sublOne}\label{subl:2.2.2}
$2.2=2.2$. \end{sublOne}

\begin{sublTwo}\label{subl:2.1.3}
$3.1=3.1$. \end{sublTwo}

Sublemmas~\refs{lem:1}{subl:2.2.1} and \refs{lem:1}{subl:2.2.2}
are sublemmas of Lemma \ref{lem:1}, whereas Sublemma~\refs{lem:2}{subl:2.1.3}
is a sublemma of Lemma \ref{lem:2}.


producing this output: 