The integer lattice is a discrete locally compact abelian group

The additive group of $\mathbb{Z}$ is abelian by [F1]. Coordinatewise addition on ${\mathbb{Z}}^d$ is associative and commutative, has the zero vector as identity, and sends $x$ to $-x$ coordinatewise as its inverse. Hence ${\mathbb{Z}}^d$ is an abelian group.

For distinct $x,y\in {\mathbb{Z}}^d$, the singleton sets $\{x\}$ and $\{y\}$ are disjoint open neighbourhoods by [F2], so the space is Hausdorff by [F6]. Each singleton $\{x\}$ is finite and hence compact by [F7]; it is an open neighbourhood of $x$ by [F2]. Thus the space is locally compact by [F8].

The product topology on ${\mathbb{Z}}^d\times {\mathbb{Z}}^d$ is discrete: for each pair $(x,y)$, the basic open set $\{x\}\times \{y\}$ is its singleton by [F2, F3]. Therefore coordinatewise addition ${\mathbb{Z}}^d\times {\mathbb{Z}}^d\to {\mathbb{Z}}^d$ and inversion ${\mathbb{Z}}^d\to {\mathbb{Z}}^d$ are continuous by [F4]. By [F5] the group is a topological group, and with step 1.2 it is locally compact Hausdorff abelian. Every map from ${\mathbb{Z}}^d$ is continuous by [F2, F4].