Proving that the utility is concave

Consider a household which solves the following problem:
$$v(k,r,w)=underset{c,lin B{(k,r,ω)}}{ {max}} {u(c,l)}$$

where $u : R_+^2 rightarrow R$ is a strictly concave, twice continuously differentiable, strictly increasing function in its two arguments: consumption, $c$, and leisure, $l$. The constraints the household must obey in selecting $c, l$ are summarized by $B$:
$$B(k, r, w) = {{c, l : 0 ≤ c ≤ rk + w(1 − l), 0 ≤ l ≤ 1}}$$

Here, $k, r, w > 0$ are numbers over which the household has no control.
Prove that $v$ is concave in $k$ and that the derivative of $v$ with respect
to $k$ exists for ‘interior $k$’. Display a formula for the derivative of $v$.

What I was thinking for solving this is by following Benveniste & Scheinkman theorem on differentiability $ω : D → R$ defined on the neighborhood $D$ of $x_0$, i.e.
$D ⊂ X$ and $x_0 ∈ int(D)$ such that: $ω(x) ≤ v(x)$ and $ω(x_0) = v(x_0)$.

And $ω(.)$ is concave and differentiable, then $v(.)$ is differentiable at $x_0$ and $v'(x_0) =ω'(x_0)$.

I’m guessing we have to substitute c with $rk + w(1 − l)$ in the utility function, but I’m confused a bit with leisure. Because after that, I think we should just get the envelope theorem or not?