Let's will write $K\_n$ for the Eilenberg-MacLane space $K(\mathbb{Z},n)$. I remind that $K\_n$ is equivalent to the loop space of $K\_{n+1}$.


Let’s consider the map $\smallsmile:K\_n\times K\_m \to K\_{n+m}$ corresponding to the cup product.


Given two elements $x:K\_n$ and $y:K\_m$, we can see $x$ as a loop in $K\_{n+1}$, then pair it with $y$ to obtain a loop in $K\_{n+1}\times K\_m$, apply the cup product there (we obtain a loop in $K\_{n+m+1}$), and finally see that loop as an element of $K\_{n+m}$. It turns out that what we obtain is equivalent to the cup product of $x$ and $y$ (there might be a sign depending on how things are set up).


Is there a reference for this result? It seems rather basic but I couldn’t find it discussed anywhere.



Formula relating the cup product in dimensions n and n+1
It is a bit of a roundabout way of proving it, but what you're looking for is immediately implied by the fact that the Eilenberg-Mac Lane spectrum is a commutative ring spectrum, as proven for example in in Example 1.14 of Stefan Schwede's [book on symmetric spectra](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwi2sKe5tpPMAhVKeT4KHZkoDuAQFggmMAE&url=http%3A%2F%2Fwww.math.uni-bonn.de%2Fpeople%2Fschwede%2FSymSpec-v3.pdf&usg=AFQjCNE9lrx6WJ1SyKsdNTSNOuCfn7kyoA&sig2=DHwovrXTKRvS-QbXIOkSuA) (note that $A[S^n]$ in that section is an explicit model for a $K(A,n)$).


