I was watching a video of an algebra lecture on line. The substitute teacher was presenting a pf. of a lemma:
$m\mathbb{Z} + n\mathbb{Z} = \mbox{gcd}(m,n)\mathbb{Z}\quad\mbox{where }m,n\in\mathbb{Z}$
I am a bit puzzled by the development of the proof in that the first assertion is that the LHS is a subgroup and has the form $d\,\mathbb{Z}$. And then goes on to use this to complete the proof.
My question (probably naive) is how does one know at the onset that the LHS is a subgroup - after all, the lemma is intended to prove what that sum is, and knowing that, it can then be claimed that the sum is also a subgroup.
In this regard, there was also a problem I saw in "Dummit and Foote" which asks to prove that the intersection of two subgroups is a subgroup. So if this is applicable to my question, it seems it has to first be determined what the intersection of $m\mathbb{Z}$ and $n\mathbb{Z}$ are. Which then leads to the gcd of the RHS.
Thanks from a self-studier.