Here is a lemma that I find myself googling regularly since I always forget it’s exact form.
Lemma 1 Let be a monotone operator, and denote by the resolvent of . Then it holds that
Proof: We start with the left hand side and deduce
I do not know any official name of this, but would call it Moreau’s identity which is the name of the respective statement for proximal operators for convex functions and :