Natural matrix norm of an inverse matrix

Question

Let $\left\|\cdot\right\| : \text{GL}(n,\mathbb{R})\to\mathbb{R}_{\ge 0}$ denote the natural matrix norm, i.e. $$\left\|A\right\|:=\max_{x\ne 0}\frac{\left\|Ax\right\|}{\left\|x\right\|}=\max_{\left\|x\right\|=1}\left\|Ax\right\|$$ is induced by a vector norm $\left\|\cdot\right\| : \mathbb{R}^n\to\mathbb{R}_{\ge 0}$. I want to show, that it holds $$\left\|A^{-1}\right\|=\left(\min_{\left\|x\right\|=1}\left\|Ax\right\|\right)^{-1}$$ Proof: \begin{equation} \begin{split} \left\|A^{-1}\right\|&=\max_{\left\|x\right\|=1}\left\|A^{-1}x\right\|\\ &=\max_{\left\|Ay\right\|=1}\left\|y\right\|\\ &=\left(\min_{\left\|Ay\right\|=1}\left\|y\right\|^{-1}\right)^{-1}\\ &=\left(\min_{\left\|x\right\|=1}\left\|Ax\right\|\right)^{-1} \end{split} \end{equation} How does the last step work?