# Use of subequations and theorem environment

I have a rather tricky (at least for me) equation numbering problem. I have a document with lots of numbered equations. Part way through the document I have a lemma (see below) for which I want to be able to reference each case with the format Lemma#(case number in lower-case roman numerals). So in my MWE I would want to reference the first case as 'Lemma 1(i)'. I have seen promise using the subequation package but wasn't able to fully get it working. How can this be done?

Here is a MWE:

\documentclass[11pt]{article}

\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathrsfs,mathtools}
\newtheorem{lemma}{Lemma}
\begin{document}

Here's a numbered equation
$$E=mc^2$$

\begin{lemma}[Explicit forms for $f(x)$]
\label{lem:forms_of_f}
$f(x)=% \begin{cases} 1 &(\mathrm i)\\[0.5em] \int_0^\infty e^{-x}\,\mathrm dx &(\mathrm{ii})\\[0.5em] 9\sum_{k=1}^\infty (10)^{-k} &(\mathrm{iii}) \end{cases}$
\end{lemma}

As you can see from Lemma \ref{reference case 1 as '1(i)'}, $f(x)=1$. However, Lemma \ref{reference case 2 as '1(ii)'}-\ref{reference case 3 as '1(iii)'} also show that $f(x)=1$. Here's another numbered equation
$$e^{\pi\mathrm i}=-1.$$

\end{document}

You can number a cases environment using numcases instead (from the cases package). Then you just have to adjust the formatting of the equation counter and store/restore it before/after the numcases environment.

\documentclass{article}

\usepackage{amsthm}
\usepackage{mathtools,cases}

\newtheorem{lemma}{Lemma}

\newcounter{saveeqncntr}

\begin{document}

Here's a numbered equation
$$E = mc^2$$

\begin{lemma}[Explicit forms for $f(x)$]
\label{lem:forms_of_f}
\begingroup
\setcounter{saveeqncntr}{\value{equation}}
\setcounter{equation}{0}
\renewcommand{\theequation}{\roman{equation}}
\begin{numcases}{f(x) =}
1                                 \label{eqn:forms_of_f_i}  \\
\int_0^\infty e^{-x}\,\mathrm{d}x \label{eqn:forms_of_f_ii} \\
9\sum_{k = 1}^\infty (10)^{-k}    \label{eqn:forms_of_f_iii}
\end{numcases}
\setcounter{equation}{\value{saveeqncntr}}
\endgroup
\end{lemma}

As you can see from Lemma~\ref{lem:forms_of_f}\eqref{eqn:forms_of_f_i}, $f(x) = 1$.
However, Lemma \ref{lem:forms_of_f}\eqref{eqn:forms_of_f_ii}--\ref{lem:forms_of_f}\eqref{eqn:forms_of_f_iii}
also show that $f(x) = 1$. Here's another numbered equation
$$e^{\pi\mathrm{i}} = -1.$$

\end{document}

I suggest an extraequations environment and to use double references.

\documentclass{article}
\usepackage{amsmath,mathtools,empheq}

\newtheorem{lemma}{Lemma}

\newenvironment{extraequations}[1][\roman]
{%
\setcounter{parentequation}{\value{equation}}
\setcounter{equation}{0}%
\renewcommand{\theequation}{#1{equation}}%
\ignorespaces
}
{%
\setcounter{equation}{\value{parentequation}}%
\ignorespacesafterend
}

\begin{document}

Here's a numbered equation
$$E=mc^2$$

\begin{extraequations}
\begin{lemma}[Explicit forms for $f(x)$]\label{lem:forms_of_f}
Text before the display, there should always be some
\begin{empheq}[left={f(x)=\empheqlbrace}]{align}
& 1 \label{case-i} \\[1ex]
& \int_0^\infty e^{-x}\,dx \label{case-ii} \\
& 9\sum_{k=1}^\infty (10)^{-k} \label{case-iii}
\end{empheq}
\end{lemma}
\end{extraequations}

As you can see from Lemma \ref{lem:forms_of_f}\eqref{case-i}, $f(x)=1$.
However, Lemma \ref{lem:forms_of_f}\eqref{case-ii} also show that $f(x)=1$.
Here's another numbered equation
$$e^{\pi i}=-1.$$

\end{document}

If you use hyperref, you need a further instruction:

\newenvironment{extraequations}[1][\roman]
{%
\setcounter{parentequation}{\value{equation}}
\setcounter{equation}{0}%
\renewcommand{\theequation}{#1{equation}}%
\renewcommand{\theHequation}{\theparentequation--#1{equation}}% to keep hyperref happy
\ignorespaces
}
{%
\setcounter{equation}{\value{parentequation}}%
\ignorespacesafterend
}

You can use \begin{extraequations}[\alph] if you want extra equations numbered by letters.