How to prove knowledge of discrete logarithm in a product? Definitions

Suppose I have two large safe primes $p$ and $q$, and a composite number $N=pq$. I have $G$, a large cyclic subgroup of $\mathbb{Z}^{*}_{N}$; $g$ and $h$ are generators of $G$. I commit to $x$ using a random value $r\in{_{R}\mathbb{Z}_{N}}$ as $(x,r)=g^{x}h^{r}\mod{N}$.

I have a set of valid messages $S=\{p_{1},p_{2},...,p_{m}\}$, and a message $x\in{S}$. The set $S$ is public, while $x$ is private.

Problem

I must prove knowledge of two secret integers $\mu$ and $\nu$: $$\mu=(\prod^{m}_{j=1}{p_{j}})/x$$ $$\nu=\mu r$$ using the commitment: $$C=g^{\prod^{m}_{j=1}{p_{j}}}h^{\nu}$$ I know this supposedly can be done using a combination of two Schnorr signatures. I just need a little guidance as to how to utilize those two signatures to prove knowledge of $\mu$ and $\nu$.

What I have so far

Given the definition of $\mu$ I have $C=g^{\mu x}h^{\nu}$. I also know I can express $g^{\mu x}=h^{-\nu}C$ and $h^{\nu}=g^{-(\mu x)}C$, so that gives me $$\mu x=\log_{g}(h^{-\nu}C)=\log_{g}(h^{-1})\log_{g}(\nu)+\log_{g}(C)$$ $$\nu=\log_{h}(g^{-(\mu x)}C)=\log_{h}(g^{-1})(\log_{h}(\mu)+\log_{h}(x))+\log_{h}(C)$$ I'm really not sure where to go from here, because I seem to be making it more complicated instead of less...
$$(x,r) = g^x h^r$$
$$C = (x,r)^ \mu$$

Now you have two standard proofs of knowledge.