How can I nest alignment environments together and get different equations numbering for the nested alignments?

Question

In the equation above, \Delta_{X{lm}} is taken as one equation by using split environment. I want to align the ranges shown in Eq. 1 which I already did, but I want each range to get its equation numbering as well, that is [\Xi \Theta] should be (1) and [\Theta \Xi] should be (2).

My source code

\documentclass[]{IEEEtran}

\usepackage{amsmath}
\usepackage{amssymb, amsfonts}

\DeclareRobustCommand{\chiup}{{\mathpalette\irchi\relax}}
\newcommand{\irchi}[2]{\raisebox{\depth}{$#1\chi$}}

\begin{document}


\begin{align}
    \begin{split}
        \Delta{X_{lm}} &= \left \{
            \begin{aligned}
                &\left[ \thinspace \Xi \quad \Theta \thinspace \right], &\chiup > 0
                \\
                &\left[ \thinspace \Theta \quad \Xi \thinspace \right], &\chiup < 0
            \end{aligned}
        \right.
    \end{split}
\label{eq:del_Xlm_bnd}
\\
\intertext{where\vspace{0.3\baselineskip}}
\Xi &= \frac{\chiup}{X_{th,kp}}
\label{eq:eq:del_Xlm_bnd_Xi}
\\
\Theta &= \frac{\chiup}{X_{th,kp} - X_{b}}
\label{eq:eq:del_Xlm_bnd_Theta}
\end{align}

\end{document}

Answer 1

Since you mentioned in the comments that numcases from the cases package cannot be nested in align, this answer by @PeterGrill won’t work. Since you also want \intertext so that you can insert text without disrupting alignment points, this answer by @egreg won’t work either.

So here is a variant of this answer by @Werner which spares you from trial and error with @Bernard’s answer.

\documentclass{IEEEtran}

\usepackage{mathtools}% Loads amsmath internally
%\usepackage{amssymb}

\DeclareRobustCommand{\chiup}{{\mathpalette\irchi\relax}}
\newcommand{\irchi}[2]{\raisebox{\depth}{$#1\chi$}}

\begin{document}

\newsavebox{\mycases}% Store the equal sign and the case brace
\sbox{\mycases}{$\displaystyle {}=\left\{\begin{array}{@{}l@{}}\mathstrut\\\mathstrut\end{array}\right.\kern-\nulldelimiterspace$}
\begin{align}
\raisebox{-.5\ht\mycases}[0pt][0pt]{$\Delta{X_{lm}}$}
  & \raisebox{-.5\ht\mycases}[0pt][0pt]{\usebox{\mycases}}
    [ \, \Xi \quad \Theta \, ], \quad \chiup > 0 \label{positive} \\
  & \hspace{\wd\mycases}
    [ \, \Theta \quad \Xi \, ], \quad \chiup < 0 \label{negative} \\
\shortintertext{where}
\Xi & = \frac{\chiup}{X_{th,kp}} \label{eq:eq:del_Xlm_bnd_Xi} \\
\Theta & = \frac{\chiup}{X_{th,kp} - X_{b}} \label{eq:eq:del_Xlm_bnd_Theta}
\end{align}

\end{document}

Additional notes:

• amsfonts is not need if amssymb is already loaded.
• The IEEEtran class tries to use newtxmath (or other Times math fonts) by default, so you shouldn’t load amssymb anyway, if Times-like fonts are used.
• Do not nest aligned within split within align. You can just use cases to write your cases (this is, of course, unrelated to your question).
• Do not manually insert vertical space as in \intertext{where\vspace{0.3\baselineskip}}. Consider using \shortintertext from the mathtools package.
• What’s the purpose of \chiup??? It seems like you somewhat “dislike” how the symbol χ sits on its natural baseline. Please don’t do this! Typographically speaking, shifting it up is simply wrong!

Answer 2

I can propose this, using the numcases and fleqn environments, from the cases and nccmath respectively – and some trial and error:

\documentclass[]{IEEEtran}

\usepackage{amsmath, amssymb}
\usepackage{cases, nccmath}
\DeclareRobustCommand{\chiup}{{\mathpalette\irchi\relax}}

\newcommand{\irchi}[2]{\raisebox{\depth}{$#1\chi$}}

\begin{document}

 \begin{numcases}{ \Delta X_{lm} = }
              \left[ \thinspace \Xi \quad \Theta \thinspace \right], &$ \chiup > 0 $
\label{eq:del_Xlm_bnd}
                \\
              \left[ \thinspace \Theta \quad \Xi \thinspace \right], &$ \chiup < 0 $
    \end{numcases}
\vspace{-1ex}
\begin{fleqn}[4.17em]
\begin{align}
  \text{where}\quad
 \Xi &= \frac{\chiup}{X_{th,kp}} & & & &
\label{eq:eq:del_Xlm_bnd_Xi}
\\
 \Theta &= \frac{\chiup}{X_{th,kp} - X_{b}}
\label{eq:eq:del_Xlm_bnd_Theta}
\end{align}
\end{fleqn}

\end{document} 

Answer 3

I am not sur I totally understood the OP's demand, but you can achieve the numbering and the alignment using empheq and align. YOu can also achieve 1.a and 1.b numbering with subequations

[![\documentclass\[\]{IEEEtran}

\usepackage{mathtools}
\usepackage{empheq}
\DeclareRobustCommand{\chiup}{{\mathpalette\irchi\relax}}
\newcommand{\irchi}\[2\]{\raisebox{\depth}{$#1\chi$}}

\begin{document}

\begin{empheq}\[left={\Delta{X_{lm}}=\empheqlbrace}\]{align}
                \left\[ \thinspace \Xi \quad \Theta \thinspace \right\], &\chiup > 0 \\
                \left\[ \thinspace \Theta \quad \Xi \thinspace \right\], &\chiup < 0 \label{eq:del_Xlm_bnd}
\end{empheq}
or
\begin{subequations}
\begin{empheq}\[left={\Delta{X_{lm}}=\empheqlbrace}\]{align}
                \left\[ \thinspace \Xi \quad \Theta \thinspace \right\], &\chiup > 0 \\
                \left\[ \thinspace \Theta \quad \Xi \thinspace \right\], &\chiup < 0 \label{eq:del_Xlm_bnd}
\end{empheq}
\end{subequations}
where
\begin{align}
\Xi &= \frac{\chiup}{X_{th,kp}} \label{eq:eq:del_Xlm_bnd_Xi}\\
\Theta &= \frac{\chiup}{X_{th,kp} - X_{b}} \label{eq:eq:del_Xlm_bnd_Theta}
\end{align}

\end{document}