Criteria for Invertibility of a Square Real Matrix

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $n\in\mathbb{N}$ and let $A\in\mathcal{M}_{n}(\mathbb{R})$ be a square real matrix, with the matrix-vector product, the origin $0$ of $\mathbb{R}^{n}$, the matrix product and the identity matrix $I_{n}$. The following conditions are equivalent.

¶ (a) The only $x\in\mathbb{R}^{n}$ with $Ax=0$ is $x=0$.

¶ (b) For $x,y\in\mathbb{R}^{n}$, $Ax=Ay$ implies $x=y$.

¶ (c) For every $b\in\mathbb{R}^{n}$ there is $x\in\mathbb{R}^{n}$ with $Ax=b$.

¶ (d) $AB=I_{n}$ for some $B\in\mathcal{M}_{n}(\mathbb{R})$.

¶ (e) $CA=I_{n}$ for some $C\in\mathcal{M}_{n}(\mathbb{R})$.

¶ (f) $A$ is invertible.