# Transversality of two mappings and diagonal

I've never taken differential topology and am confused by the definition of transversality, and while trying to solve the following I got stuck.

Given smooth manifolds and maps $$f:M\to N$$ and $$g:P\to N$$, show that $$f$$ and $$g$$ are transversal to each other if and only if $$f\times g: M\times P\to N\times N$$ is transversal to the diagonal $$\Delta \subset N\times N$$.

By definition $$f \pitchfork g$$ means that for every $$x \in M$$ and $$y \in P$$ such that $$z\doteq f(x) = g(y)$$, we have $${\rm d}f_x[T_xM] + {\rm d}g_y[T_yP] = T_zN.$$This is not required to be a direct sum. If $$S \subseteq N$$ is a submanifold, we write $$f \pitchfork S$$ instead to mean $$f \pitchfork \iota_S$$, where $$\iota_S\colon S \hookrightarrow N$$ is the inclusion map.
It is easy to see that $${\rm d}(f\times g)_{(x,y)}[M\times P] = {\rm d}f_x[T_xM] \times {\rm d}g_y[T_yP]$$. Also, we have that the tangent space to the diagonal is the diagonal of the tangent space, i.e., $$T_{(z,z)}\Delta = \{ (w,w) \mid w \in T_zN\}$$.
$$\implies:$$ Assume that $$f \pitchfork g$$, and let's prove that $$(f\times g) \pitchfork \Delta$$. Assume that $$(x,y) \in M\times P$$ are such that $$z \doteq f(x) = g(y)$$ and take $$(w_1,w_2) \in T_{(z,z)}(N\times N) = T_zN\times T_zN$$. Our goal is to write $$(w_1,w_2)$$ as the sum of something in $${\rm d}f_x[T_xM]\times {\rm d}g_y[T_yP]$$ with something in $$T_{(z,z)}\Delta$$. Well, $$f \pitchfork g$$ gives $$u \in T_xM$$ and $$v\in T_yP$$ such that $$w_1-w_2 = {\rm d}f_x(u) - {\rm d}g_y(v)$$. Now let $$w = w_1 - {\rm d}f_x(u)$$ and note that $$w = w_2 - {\rm d}g_y(v)$$ holds as well. Then we have that $$(w_1,w_2) = ({\rm d}f_x(u),{\rm d}g_y(v)) + (w,w),$$and this shows that $$(f\times g)\pitchfork \Delta$$, as wanted.
$$\impliedby:$$ Assume that $$(f\times g)\pitchfork \Delta$$ and let's show that $$f \pitchfork g$$. So, take $$w \in T_zN$$ and look at $$(w,0) \in T_{(z,z)}(N\times N)$$. The assumption $$(f\times g)\pitchfork \Delta$$ provides $$u \in T_xM$$, $$v \in T_yP$$ and $$w' \in T_zN$$ such that $$(w,0) = ({\rm d}f_x(u), {\rm d}g_y(v)) + (w',w').$$It readily follows that $$w = {\rm d}f_x(u)+{\rm d}g_y(-v)$$, which shows that $$f \pitchfork g$$.