Theorem referencing from previous section

Question

I have my theorems labeled as subsection.theoremnumber. So for example: Theorem 2.3 is the third theorem in subsection 2. If the subsection is in say, section 4, I would like when I reference this theorem in say, section 6, for it to read Theorem 4.2.3.

I know that this is similar to refer to theorems in previous chapter, but the answers there do not work for sections.

Answer 1

By changing the word "chapter" to "section" everywhere in the code which is linked in my question, I achieved my desired result.

Output:

Code:

\documentclass{amsart} 
\usepackage{amsthm}
\usepackage{zref}% http://ctan.org/pkg/zref

\makeatletter
\let\oldlabel\label
\renewcommand{\label}[1]{%
\zref@labelbylist{#1}{special}% Special label
\oldlabel{#1}% Old label
}
\newcounter{splabel}
\zref@newlist{special}% Create a new property list called special
\zref@newprop{section}{\arabic{section}}% Section property holds \arabic{section}
\zref@addprop{special}{section}% Add a section property to special
\newcommand*{\thmref}[1]{%
  \stepcounter{splabel}% Increment local "special label" counter
\zref@labelbylist{#1-\thesplabel}{special}% Create label
\edef\targetsec{\zref@extractdefault{#1}{section}{-1}}% Extract target section
\edef\sourcesec{\zref@extractdefault{#1-\thesplabel}{section}{-1}}% Extract source section
\ifnum\targetsec=\sourcesec\else\targetsec.\fi%
\ref{#1}%
}

\newtheorem{dummy}{***}[subsection]% Used so that theorems, definitions, etc can have same counter within subsections
\newtheorem{theorem}[dummy]{Theorem}
\newtheorem{lemma}[dummy]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[dummy]{Definition}
\newtheorem{remark}[dummy]{Remark}

\renewcommand{\thesubsection}{\arabic{subsection}}% Custom numbering on subsection to remove section number

\begin{document}

\section{First Section}
\subsection{First Subsection in Section 1}
\begin{definition} 
    Good definition.
\end{definition}

\subsection{Second Subsection in Section 1}
\begin{remark}
    Interesting remark. 
\end{remark}
\begin{lemma} \label{lem:name}
    Useful lemma.
\end{lemma}

\subsection{Third subsection in Section 1}

\begin{theorem} \label{thm:name}
    Important theorem.
\end{theorem}

\section{Section 2}
\subsection{Another subsection} There's not much to say here. 
\subsection{Last Subsection}\ 
 As you can see, Theorem \thmref{thm:name}  is very important. Indeed, Lemma~\thmref{lem:name} was quite useful. 
\end{document}