Can't understand a proof: Let $a,b,c$ be integers. If $a$ and $b$ divide $c$, then $lcm(a,b)$ also divides $c$

It follows from the (general) fact that if $a$ divides $b$ and $c$, then $a$ also divides $b-c$, so in this case:

$c = q.lcm(a,b) + r$ and $0 \leq r < lcm(a,b)$

$c - q.lcm(a,b) = r$

Which means that both $a$ and $b$ divides $r$, but $0 \leq r < lcm(a,b)$. So $r$ must be equal to $0$, and we get:

$c = q.lcm(a,b)$, as you wish.

The statement needs no proof, given that you define things properly. See also: partially ordered set, lattice, semilattice, join, meet.

Definition 0. Given $a,b \in \mathbb{N},$ lets say that $a$ divides $b$ iff there exists $k \in \mathbb{N}$ with $ka=b$. We denote this $a \mid b$.

Exercise: Show that $\mid$ is a partial order on $\mathbb{N},$ and verify that $1$ is the least element and $0$ is the greatest element with respect to this order.

Definition 1. Given $a,b \in \mathbb{N},$ the lowest common multiple of $a$ and $b$, denoted $\mathrm{lcm}(a,b),$ is the unique natural number such that for a generic natural number $n$, we have the following equivalence.

$$(a \mid n) \wedge (b \mid n) \leftrightarrow \mathrm{lcm}(a,b) \mid n$$

Exercise: Show that $\mathrm{lcm}$ is a well-defined function. (Technically, what' you're trying to do is to show that binary joins exist in the poset of natural numbers ordered by division.)