Proving that the language of satifiable CNF formulae with primes is NP-complete Given the following language:
  
  $$L=\left\{\langle\phi, n\rangle \ \middle|\  \begin{array}{l}\phi\text{ is a satisfiable Boolean formula}\\ \text{written as POS (in CNF form)}\\ \text{and $n$ is a prime number}\end{array}\right\},$$
  
  Prove that L is an NPC language.


This is the first time I'm trying to solve this kind of questions, and I am stuck.

This is what I have so far:

To prove that a language is NPC, I need to prove two things:

1) $L \in \mathrm{NP}$

2) For any $L'\in \mathrm{NP}$, there exists a poly-time reduction from $L'$ to $L$.

To prove (1), can I say that $\phi$ is NP (because we know that SAT is NP), and that $n$ is P (because Prime is P)?

If so, I think that I need to say that $\mathrm{P} \subseteq \mathrm{NP}$ to complete the proof of (1).

What's next?
For 1): You have to prove that the language is solvable by a nondeterministic TM in polynomial time. The easiest way is indeed to use previously defined TMs, e.g. the TM to solve SAT. So your approach is correct.

For 2): You don't have to prove it for every $L'$ $\in$ $NP$ explicitly, it suffices to give a reduction from a language that is NP-complete. The idea is that you reduce every word $w' \in L'$ to a word $w \in L$, so if you find an efficient way to solve $L$, you also have an efficient way to solve $L'$. So you can choose any NP-complete language that should provide a simple reduction to $L$.

The next step therefore would be to choose such NP-complete language $L'$.

Next you have to find a reduction $f$ to reduce your previously chosen language $L'$ to your language $L$. Depending on your choice of $L'$, this can be a very simple process in your case. Note that $\{$ $f(w') \mid w' \in L \} \subseteq L$, not $\{$ $f(w') \mid w' \in L \} = L$