Let $K$ be a complete local division ring (note $v$ its valuation). For $x,y\in K$ ($y\ne0$), one puts $x^y=yxy^{-1}$. Let $r\in\mathbb N$. Consider $x,y\in K$ and $a,b\in K^*$ such that $v(x-y)\ge r$ and $v(a-b)\ge r$. Do we have $v(x^a-y^b)\ge r$? In the commutative case, it is obvious but in the non-commutative case, I can not see the answer.

Thanks in advance