@modulux A proof is not information in a strict sense, and largery exactly because of this reason: it's self-contained (or, well, can be, with sufficient formalism available).
In a broad sense, there's some very interesting philosophy that can be done about the notion of information content of Teh Book. But it's mostly the kind of philosophy that requires a larger mug of beer than would be conducive to my upcoming meetings[1], so, as the old Orcish saying goes, nar udautas.
As a general rule, I tend to prefer the interpretation that a proof is a series of "I'd now like to bring your attention to ..." kind of steps: they don't add anything (directly) to your mental map; they suggest where you should look at to find interesting things that are already on the map.
[1] A children's book I once read included a character, one mathematics professor, who argued that it is pointless to ask questions, because there's two possibilities: the answer either is known or is not known. If it's known, what's the point of asking it again? If it's not known, what's the point of asking if there won't be an answer?
And, well, while it's silly in an obvious way, this kind of reasoning actually comes up in the context of proofs-as-information.
