To prove your pdf existed, I would go a similar route as you took: I would hash the doc, convert the private key to an address and verify that the address has received money in the past. If it has, either the document must have existed or the extremely unlikely event that someone chose the hash of the document as a private key by accident must have happened. Since we can rule out the latter with high probability, the documents existence at the time of the transaction is proven.

Please note: you should move the money from that address. Anyone with the document can access it.