I am reading the paper "Proximal algorithms" by N. Parikh and S. Boyd, and I found interesting the basic properties of proximal operators. However, I can't prove the equivalence for the mentioned properties (except the first one).
2.2 Basic Operations
This section can be referred to as needed; these properties will not play a central role in the rest of the paper.
Postcomposition: If $f(x) = \alpha\phi(x) + b$, with $\alpha > 0$, then $$\text{prox}_{\lambda f}(v) = \text{prox}_{\alpha \lambda \phi}(v).$$
Precomposition: If $f(x) = \phi(\alpha x + b)$, with $\alpha \neq 0$, then $$\text{prox}_{\lambda f}(v) = \frac{1}{\alpha} \left( \text{prox}_{\alpha^2 \lambda \phi}(\alpha v + b) - b \right).$$
Orthogonal Precomposition: If $f(x) = \phi(Qx)$, where $Q$ is orthogonal $Q^TQ = QQ^T = I $, then $$\text{prox}_{\lambda f}(v) = Q^T \text{prox}_{\lambda \phi}(Qv).$$
Affine Addition: If $f(x) = \phi(x) + a^T x + b$, then $$ \text{prox}_{\lambda f}(v) = \text{prox}_{\lambda \phi}(v - \lambda a).$$
Regularization: If $f(x) = \phi(x) + \frac{\rho}{2} \|x - a\|^2_2$, then $$\text{prox}_{\lambda f}(v) = \text{prox}_{\tilde{\lambda} \phi} \left( \frac{\tilde{\lambda}}{\lambda}v + \frac{\rho \tilde{\lambda}}{\lambda} a \right),$$ where $\tilde{\lambda} = \frac{\lambda}{1 + \lambda \rho}$.