Let be $M := M^m$ and $N := N^n$ smooth manifolds and $\pi_1: M \times N \longrightarrow M$, $\pi_2: M \times N \longrightarrow N$ the canonical projections into the factors.
a) Show that $\pi_1$ and $\pi_2$ are smooth maps.
b) Let be a smooth manifold $P$ and $f: P \longrightarrow M \times N$ a map. Show that $f$ is smooth if and only if $\pi_i \circ f$ for $i = 1,2$.
c) Show that the injections $i_p: N \longrightarrow M \times N$, $j_q: M \longrightarrow M \times N$ given by $i_p(q) = (p,q)$ and $j_q(p) = (p,q)$ are smooth.
I would like to know if my attempt is correct.
$\textbf{My attempt:}$
a) Let be $\{ (U_{\alpha},X_{\alpha}) \}$ and $\{ (V_{\beta},Y_{\beta}) \}$ smooth atlas of $M$, and $N$, respectively, $p \in M$, $q \in N$, $(x_1, \cdots, x_m) := X_{\alpha}(p)$, $(x_{m+1}, \cdots, x_{m+n}) := Y_{\beta}(q)$ and let be $\{ (U_{\alpha} \times V_{\beta}, (X_{\alpha},Y_{\beta})) \}$ a smooth atlas of $M \times N$, where $(X_{\alpha},Y_{\beta})(p,q) := (X_{\alpha}(p),Y_{\beta}(q))$. Thus,
$$X_{\alpha} \circ \pi_1 \circ (X_{\alpha},Y_{\beta})^{-1}: Y_{\beta}(V_{\beta}) \longrightarrow X_{\alpha}(U_{\alpha})$$
and
$$\begin{align*} (X_{\alpha} \circ \pi_1 \circ (X_{\alpha},Y_{\beta})^{-1})(x_1, \cdots, x_m, x_{m+1}, \cdots, x_{m+n}) &= X_{\alpha}(\pi_1((X_{\alpha},Y_{\beta})^{-1}(x_1, \cdots, x_m, x_{m+1}, \cdots, x_{m+n})))\\ &= X_{\alpha}(\pi_1(p,q)) = X_{\alpha}(p) = (x_1, \cdots, x_m), \end{align*}$$ which is simply the projection of $X_{\alpha}(U_{\alpha}) \times Y_{\beta}(V_{\beta}) \subset \mathbb{R}^{m+n}$ into $X_{\alpha}(U_{\alpha}) \subset \mathbb{R^m}$, therefore $X_{\alpha} \circ \pi_1 \circ (X_{\alpha},Y_{\beta})^{-1})$ is smooth and, consequently, $\pi_1$ is a smooth map.
Analogously, $\pi_2$ is a smooth map.
b)
If $f$ is a smooth map, then $\pi_i \circ f$ is a smooth map for $i = 1,2$ by item a.
Conversely, let be $\{ (W_{\gamma},Z_{\gamma}) \}$ a smooth atlas of $P$, then $X_{\alpha} \circ f \circ Z_{\gamma}^{-1}$ and $Y_{\beta} \circ f \circ Z_{\gamma}^{-1}$ are differentiable in the sense of analysis on Euclidean spaces by hypothesis, therefore $(X_{\alpha},Y_{\beta}) \circ f \circ Z_{\gamma}^{-1}$ is differentiable in the sense of analysis on Euclidean spaces by a result of analysis on Euclidean spaces, then $f$ is a smooth map.
c)
It's just observe that $\pi_1 \circ i_p$ is the constant map ($(\pi_1 \circ i_p)(q) = p$ for all $q \in N$) and $\pi_2 \circ i_p \equiv id_N$, where $id_N$ is the identity in $N$, then $i_p$ is a smooth map by the item b. Analogously, $j_q$ is a smooth map.
Thanks in advance!