Differential geometry EDP

Question

I'm reading the article "A compactness theorem for complete Ricci shrinkers" https://arxiv.org/pdf/1005.3255.pdf On page 9 they say that from $$\sup_{B_r(p)} |Rm_{g}| \leq C_0 (r),\quad \sup_{B_r(p)} |f| \leq C_0 (r), \quad \sup_{B_r(p)} |\nabla f| \leq C_1 (r) $$ and $$\triangle Rm = \nabla f * \nabla Rm + Rm + Rm*Rm, \quad \triangle f = \frac{n}{2} - R, $$ you can get that $$\sup_{B_r(p)}|\nabla^{\ell}Rm_{g}| \leq C_{\ell} (r),\quad and \quad \sup_{B_r(p)} |\nabla ^ {\ell} f| \leq \tilde{C}_{\ell} (r)$$ where $Rm$ is the curvature tensor. They say that this follows "by some very well-know arguments". Is there a standard technique for this type of problem, where is involved an elliptic equation?.