Tangent space of a Product of two manifolds

Question

Suppose $M$ and $N$ are two $C^\infty$ manifolds. Take $p\in M$ and $q\in N$. We have the following maps between these: $\iota_1 : M\to M\times N$, $\iota_2:N\to M\times N$, $\pi_1:M\times N\to M$ and $\pi_2:M\times N\to N$ defined by: $\iota_1(x)=(x,q)$, $\iota_2(y)=(p,y)$, $\pi_1(x,y)=x$, and $\pi_2(x,y)=y$. I want to prove that $d(\iota_1\circ\pi_1)_{(p,q)}+d(\iota_2\circ\pi_2)_{(p,q)}=\text{Id}_ {T_{(p,q)}(M\times N)}$. I can see that this must be the case using the fact that $T_{(p,q)}(M\times N)\cong T_{p}(M)\times T_q(N)$ in a natural way. But that proof involves a dimension argument. Is there a way to prove this assertion directly? (Here, by a notation such as $dF_x$ I mean the tangent map or pushforward of $F$ at $x$.)