If i have two independent log normal distributions, where;

Y1~LN(Mu1,Sigma1^2) and Y2~LN(Mu2, Sigma2^2), and Y=Y1*Y2

Am i right in thinking;

Y~LN(Mu1+Mu2, Sigma1^2+Sigma2^2) ?

If this is correct, how do i go about proving this? I have tried to substitute Y=exp(y) for y1 and y2 accordingly, and multiply the pdf's together due to independence, but do not seem to get anywhere near my desired answer.

If this is incorrect, is there an identity i can use?

Thanks