Suppose $(X,x)$ and $(Y,y)$ are two pointed compact manifolds. Every introduction to algebraic topology mentions the result
$$\pi(X \times Y, (x,y)) \cong \pi(X, x) \times \pi( Y, y) $$
where $\pi(X, x)$ say denotes the fundamental group with basepoint $x \in X$ et cetera.
It seems, for compact manifolds at least, a modification of the finite proof along with Tychonoff's theorem should spit out the infinite version of the above
$$\pi \left (\prod_i X_i, (x_i) \right ) \cong\prod_i \pi(X_i, x_i).$$
I would like to cite this result as part of a counterexample, ideally introducing as little machinery as possible. So far I can only find the result as an exercise. Does anyone know a good reference for the complete proof?
There is a proof of this (for arbitrary homotopy groups, not just $\pi_1$) as Proposition 4.2 in Hatcher's Algebraic Topology, if you just want a source you can cite. The proof is only two sentences long and gives very little detail, though, so this is not a good reference if you want a detailed proof to refer readers to.