Yes, you can check the completed purchase after uninstall app and install it again.
All purchased items stored on the Play Server and particularly in your installed Google Play app. So after item is purchased once the information about purchase is stored and you can request it at any time untill you "consume" this purchase. "Consume" means reset the information about purchase to allow make new purchase with same ID. This is only for v3 if Android Billing Library, because all purchases in v3 are one-off (you can buy it only once) and have to be "consumed" to be purchased again.
The identifier of the purchase item is a simple string that you define in the Google Developer Console. You should login to the developer console, create new project for the future app, upload signed .apk (necessary), after that you have to select the project, move to "In-App products" section and there you can create necessary purchase items with purchase IDs. While purchase action in the application you should pass the matched purchaseID to the launchPurchaseFlow() method on the IabHelper object (these are all from v3 Billing Library). The information about purchased items can also be received by using matched purchaseID.
Official Google developer documentation has pretty decent amount of information about implementing and using Billing Library: