Estimate in quantum mechanics

Question

I am reading the following paper:[on quantum information theory][1]

Does anybody understand the estimate at the bottom of page $7$

$$\left\lVert \rho-\rho_n \right\rVert \le 2 \left\lVert (id_A-P_n)\otimes id_R \rho (P_n \otimes id_R)\right\rVert_1+ 2 r_n?$$

I have troubles understanding what happened in this step. Can anybody explain this one to me?

  [1]: https://arxiv.org/pdf/1706.00361.pdf