A lemma in Gerschgorin theorem

Let :

• $$n geq 2$$
• $$A in mathcal{M}_n( mathbb{C} )$$
• $$forall i leq n , |a_{i,i}| > sum_{j ne i}^{n} |a_{i,j}|$$
• $$V in mathcal{M}_{n,1}( mathbb{C} ), V ne 0 , Y=AV$$
• $$V=(v_i)_i$$ and $$Y=(y_i)_i$$

We want to prove that :
$$forall i leq n , |a_{i,i}| |v_i| – sum_{j ne i}^{n} |a_{i,j}| |v_j| leq |y_i|$$

My attempt :
$$sum_{j ne k} a_{i,j} z_j + a_{i,k} z_k =0$$