Why does short run market supply become more elastic with more firms

I have heard a justification along the lines of this: The supply curve becomes flatter (more elastic) with more firms in the market, because a given increase in price calls forth more production when there are many firms rather than one. This however does not make sense to me as elasticity only cares about proportional change. Mathematically is there are $N$ firms with supply functions $q(p)$ and a market supply of $Q(p) = Nq(p)$ then
$$ frac{p}{Q}frac{mathrm dQ}{mathrm dp}=frac{p}{Nq}frac{mathrm dNq}{mathrm dp}=frac{p}{q}frac{mathrm dq}{mathrm dp}$$