Older computer dead, but had just renewed... Can I reinstall?

I had purchased a computer in 2007 with Norton installed.  On 1/9/09, I renewed my Norton Internet Security Subscription.  On 2/11/09, my computer was destroyed due to power surge.  I have ordered a new computer with no anti-virus/security software installed, as my only other option was McAfee, which I do not want.  I have retained my product key and serial number from my renewal on 1/8/09.  My question is, can I download the Norton Internet Security as a new product on my computer (with my product key that I saved), even though I had just purchased the Subscription renewal?  Will providing this product key install all the necessary files onto my new computer so that the computer is protected correctly?