Manipulating a linear form into a proximal operator form

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Given an expression:

$$\arg\min_x -\alpha p^TAx + f(x)$$

I'd like to define $v$ such that:

$$\text{prox}_{\sigma f}(v) = \arg\min_x 0.5 \|x - v\|_2^2 + \sigma f(x) = \arg \min_x -\alpha p^TAx + f(x)$$

I abuse equality above to mean the same argument.
How can I find such $v$?

Such trick is done in:

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