Write proof that a matrix $N\times N$ with all values greater than 5 and $\vec x,\vec y$ being stochastic then $\sum \vec xA\vec y^T\ge 5$

Question

Better way to write a proof that a matrix $N\times N$ with all values greater than 5 and $\vec x,\vec y$ being stochastic then $\sum \vec xA\vec y^T\ge 5$

I have a proof for this but it's a bit wordy

"Since all indexes in A are at least 5 then any stochastic vector because it sums to 1, by property of matrix multiplication multiplied by A must create a vector of elements $\ge$5, then since x is another stochastic vector that sums to 5, the sum of that times a vector where every element is greater or equal to 5 will be greater than or equal to 5 as well."

Is there a more elegant way to write this?

Answer 1

Let $\vec e=(1,1,\ldots,1)$. By assumption, $A\ge5\vec e^T\vec e$ entrywise. Therefore $$ \vec xA\vec y^T\ge \vec x(5\vec e^T\vec e)\vec y^T=5(\vec x\,\vec e^T)(\vec e\,\vec y^T)=5. $$