Enumerate environments within theorems

Question

How to create an environment for enumerating that allows one to identify the enumeration with the theorem environment?

Example:

---

Definition 1.1 Let X be a set. An algebra over X is a collection C of subsets of X satisfying

D 1.1.1 If A is an element of C, then X\A is an element of C;

D 1.1.2 If A and B are both elements of C, then the union A U B is an element of C.

---

Proposition 1.1 Let C be an algebra over a set X; then the following senteces are true

P 1.1.1 The empty set is an element of C;

P 1.1.2 The set X is an element of C;

P 1.1.3 Every finite union of elements of C is an element of C;

P 1.1.4 Every finite intersection of elements of C is an element of C.

---

It would be quite useful if I could labeled then, to a further reference.

Answer 1

Here's one possible solution using the enumitem package to define a new list-like environment whose label uses a variable prefix; this prefix is controlled by a macro and, with the help of the etoolbox package, the theorem-like environments are patched to redefine the prefix. Labeling and cross-referencig items is then done as usual.

According to barbara beeton's comment, provisions were made to have the labels with a final period and the cross-references without it. Also, italicized item numbers were suppressed.

\documentclass{book}
\usepackage{amsthm}
\usepackage{enumitem}
\usepackage{etoolbox}

\newtheorem{prop}{Proposition}[chapter]
\theoremstyle{definition}
\newtheorem{defi}{Definition}[chapter]

\newcommand\EnumPrefix{}

\newlist{senenum}{enumerate}{10}
\setlist[senenum]{label=\EnumPrefix.,ref=\EnumPrefix,leftmargin=*}

\AtBeginEnvironment{defi}{\renewcommand\EnumPrefix{\normalfont\bfseries D.\thedefi.\arabic*}}
\AtBeginEnvironment{prop}{\renewcommand\EnumPrefix{\normalfont\bfseries P.\theprop.\arabic*}}

\begin{document}

\chapter{Test Chapter}

\begin{defi}
Let $X$ be a set. An algebra over $X$ is a collection $C$ of subsets of $X$ satisfying
\begin{senenum}
\item If $A$ is an element of $C$, then $X\setminus A$ is an element of $C$;
\item  If $A$ and $B$ are both elements of $C$, then the union $A\cup B$ is an element of $C$.
\end{senenum}
\end{defi}

\begin{prop}
Let $C$ be an algebra over a set $X$; then the following senteces are true
\begin{senenum}
\item\label{ite:algempty} The empty set is an element of $C$;
\item The set $X$ is an element of $C$;
\item  Every finite union of elements of $C$ is an element of $C$;
\item\label{ite:alginter} Every finite intersection of elements of $C$ is an element of $C$.
\end{senenum}
\end{prop}

In the proof of the equivalence of \ref{ite:alginter} and \ref{ite:algempty}, we used,...

\end{document}

The above approach focuses on the list-like environment, therefore it is ready to use (with only minor changes) in the case in which ntheorem is used to define the theorem-like structures (ntheorem doesn't use a default period at the end of theorem numbering):

\documentclass{book}
\usepackage{ntheorem}
\usepackage{enumitem}
\usepackage{etoolbox}

\newtheorem{prop}{Proposition}[chapter]
\theoremstyle{changebreak}
\newtheorem{defi}{Definition}[chapter]

\newcommand\EnumPrefix{}

\newlist{senenum}{enumerate}{10}
\setlist[senenum]{label=\EnumPrefix,leftmargin=*}

\AtBeginEnvironment{defi}{\renewcommand\EnumPrefix{\normalfont\bfseries D.\thedefi.\arabic*}}
\AtBeginEnvironment{prop}{\renewcommand\EnumPrefix{\normalfont\bfseries P.\theprop.\arabic*}}

\begin{document}

\chapter{Test Chapter}

\begin{defi}
Let $X$ be a set. An algebra over $X$ is a collection $C$ of subsets of $X$ satisfying
\begin{senenum}
\item If $A$ is an element of $C$, then $X\setminus A$ is an element of $C$;
\item  If $A$ and $B$ are both elements of $C$, then the union $A\cup B$ is an element of $C$.
\end{senenum}
\end{defi}

\begin{prop}
Let $C$ be an algebra over a set $X$; then the following senteces are true
\begin{senenum}
\item\label{ite:algempty} The empty set is an element of $C$;
\item The set $X$ is an element of $C$;
\item  Every finite union of elements of $C$ is an element of $C$;
\item\label{ite:alginter} Every finite intersection of elements of $C$ is an element of $C$.
\end{senenum}
\end{prop}

In the proof of the equivalence of \ref{ite:alginter} and \ref{ite:algempty}, we used,...

\end{document}

When using ntheorem, there's even another option, not requiring the etoolbox package since \theoremprework can be used to redefine appropriately the prefix used for the list-like environment (as suggested by cmhughes in a comment to the original question); here's the code corresponding to this approach and producing the same result as before:

\documentclass{book}
\usepackage{ntheorem}
\usepackage{enumitem}

\newcommand\EnumPrefix{}

\theoremprework{\renewcommand\EnumPrefix{\normalfont\bfseries P.\theprop.\arabic*}}
\newtheorem{prop}{Proposition}[chapter]
\theoremstyle{changebreak}
\theoremprework{\renewcommand\EnumPrefix{\normalfont\bfseries D.\thedefi.\arabic*}}
\newtheorem{defi}{Definition}[chapter]

\newlist{senenum}{enumerate}{10}
\setlist[senenum]{label=\EnumPrefix,leftmargin=*}

\begin{document}

\chapter{Test Chapter}

\begin{defi}
Let $X$ be a set. An algebra over $X$ is a collection $C$ of subsets of $X$ satisfying
\begin{senenum}
\item If $A$ is an element of $C$, then $X\setminus A$ is an element of $C$;
\item  If $A$ and $B$ are both elements of $C$, then the union $A\cup B$ is an element of $C$.
\end{senenum}
\end{defi}

\begin{prop}
Let $C$ be an algebra over a set $X$; then the following senteces are true
\begin{senenum}
\item\label{ite:algempty} The empty set is an element of $C$;
\item The set $X$ is an element of $C$;
\item  Every finite union of elements of $C$ is an element of $C$;
\item\label{ite:alginter} Every finite intersection of elements of $C$ is an element of $C$.
\end{senenum}
\end{prop}

In the proof of the equivalence of \ref{ite:alginter} and \ref{ite:algempty}, we used,...

\end{document}