It's absolutely been demonstrated to be true, many many times. Do you think game studios, console makers, movie studios etc put money into DRM because none of them understand their own business?

There are lots of people who spoken either on or off the record who say the same things: the moment their copy protection fails, sales fall off a cliff and you can see it in the data. For video games it's even worse. The general figure I see for video game piracy on the PC platform is around 90-95% of players are not paying for it, yet, often they are still consuming resources on e.g. multiplayer servers.

Movie studio executives have graphs of sales and you can see the point at which high quality rips were uploaded to the internet.

Example:

http://www.develop-online.net/news/lgc-2013-sports-interacti...