Unboundedness of a linear program via extreme rays of the feasible set

For all $m, n \ge 0$, an $m \times n$ real matrix $A$, $b \in \mathbb{R}^m$, and $c \in \mathbb{R}^n$, assume the polyhedron $P = \{x : A x \ge b \text{ componentwise}\}$ has at least one extreme point (standard sense). Then the following are equivalent: (i) $\inf\{c \cdot x : x \in P\} = -\infty$ in the extended reals (this in particular forces $P \ne \emptyset$, since the infimum over an empty set is $+\infty$); (ii) there exists $d$ with $c \cdot d < 0$ (strict) which is an extreme ray of the matrix $A$ in this development's sense, namely: $A d \ge 0$ componentwise, $d \ne 0$, and there exists a finite set $s$ of row indices with exactly $n-1$ elements (natural-number subtraction, so $0$ elements when $n = 0$) such that $A_i \cdot d = 0$ for every $i \in s$ and the rows $(A_i)_{i \in s}$ are linearly independent over $\mathbb{R}$. Note that the extreme-ray notion in (ii) refers only to the matrix $A$ (via the cone $\{d : A d \ge 0\}$) and is independent of the right-hand side $b$.