How to refer to custom-numbered theorem?

Question

Short form: Suppose I have sporadically numbered theorems:

Thm A
Thm 1
Thm B

Code below shows how to force this numbering when printing the Theorems. However, when I refer back to these via \ref, they still show up as "Thm 1", "Thm 2", "Thm 3", respectively. How to force \ref to show their displayed titles?

History of this question

Long form: I devised a way to replicate Serre's numbering of Corollaries by the number of the preceding Lemma / Proposition [see bottom for full code]. For example:

Section 1
Lemma 1.1
Corollary 1.1
Proposition 1
Corollary 1

Section 2
Lemma 2.1
Proposition 2
Lemma 2.2
Corollary 2.2A
Corollary 2.2B
I want to say: by Corollary 2.2A, we win!
Instead, TeX gives: by Corollary 2, we win!

However, the displayed Corollary number does not affect the internal numbering of the Corollary. For example, the first replem corollary (1.1) is internally remembered as Corollary 1, the second one (2.1) as Corollary 2, etc.
Then, when I refer to the corollary in text, I get a reference to the internal number instead. How to modify the internal numbering to match the labeled text?

\documentclass[11pt]{amsart}
\usepackage{amsmath}

\makeatletter
\newtheorem*{rep@theorem}{\rep@title}
\newcommand{\newreptheorem}[2]{%
\newenvironment{rep#1}[2][]{%
 \def\rep@title{#2 \ref{##2}##1}%
 \begin{rep@theorem}}%
 {\end{rep@theorem}}}
\makeatother

\newtheorem{lem}{Lemma}
\newreptheorem{lem}{Corollary}
\newtheorem{Lem}[lem]{Lemma}%*
\numberwithin{lem}{section}
\newtheorem{prop}{Proposition}
\newreptheorem{prop}{Corollary}
\newtheorem{Prop}[prop]{Proposition}
\newtheorem{Cor}{Corollary}%}[lem]{
\newtheorem{cor}[Cor]{Corollary}%*

\begin{document}
\section{asdf}
\begin{lem}\label{lem1}
\end{lem}
\begin{replem}{lem1}\label{cor1}
\end{replem}
\begin{prop}\label{p1}
\end{prop}
\begin{repprop}{p1}\label{cor2}
\end{repprop}
\section{fdsa}
\begin{lem}\label{lem2}
\end{lem}
\begin{prop}\label{p2}
\end{prop}
\begin{lem}\label{lem3}
\end{lem}
\begin{replem}[A]{lem3}\label{cor3}
\end{replem}
\begin{replem}[B]{lem3}\label{cor4}
\end{replem}

By Corollary \ref{cor3}, we win!
\end{document}

One solution is to say: "By Corollary \ref{lem3}A..." instead of "By Corollary \ref{cor3}; however, the corollaries tends to move frequently, so I'd like a way to make the reference independent of the actual Lemma.

Answer 1

A 'trial' of a solution, by querying the state of the \@lemmamode conditional and the value of the proposition counter.

This way, the references seem to be correct.

However, this will get into trouble at the time hyperref comes into action!

\documentclass[11pt]{amsart}
\usepackage{amsmath}
\usepackage{etoolbox}

\newcommand{\corcntrfmtlem}{%
  \ifnum\value{prop} > 0\relax
  \thelem.\Alph{cor}%
  \else
  \arabic{lem}.\arabic{cor}%
  \fi
}

\newcommand{\corcntrfmtprop}{%
  \arabic{cor}%
}


\makeatletter
\newif\if@lemmamode


\newtheorem{lem}{Lemma}[section]
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}[prop]% Restart the cor counter

\AfterEndEnvironment{lem}{\@lemmamodetrue}% Enable lemma mode
\AfterEndEnvironment{prop}{\@lemmamodefalse} % Disable lemma mode

\renewcommand{\thecor}{%
  \if@lemmamode
  \corcntrfmtlem%
  \else
  \corcntrfmtprop%
  \fi\relax
}
\makeatother

\begin{document}
\section{asdf}
\begin{lem}\label{lem1}
\end{lem}
\begin{cor}\label{cor1}
\end{cor}
\begin{prop}\label{p1}
\end{prop}

\begin{cor}\label{cor2}
\end{cor}


\section{fdsa}
\begin{lem}\label{lem2}
\end{lem}
\begin{prop}\label{p2}
\end{prop}
\begin{lem}\label{lem3}
\end{lem}
\begin{cor}\label{cor2a}
\end{cor}
\begin{cor}\label{cor2b}
\end{cor}



By Corollary \ref{cor2a} and \ref{cor2b}, we will win!
\end{document}