How long do a spellcaster's prepared spells stay prepared?

Question

I realised today that an assumption I was making, that spellcasters who prepare a spell list must do so every day, might not be true in fifth edition. The rules are very clear about how long it takes a wizard to prepare an altered list of prepared spells:

You can change your list of prepared spells when you finish a long rest. Preparing a new list of wizard spells requires time spent studying your spellbook and memorizing the incantations and gestures you must make to cast the spell: at least 1 minute per spell level for each spell on your list.

However, there's no reference to spending time preparing the same list of spells prepared the previous day. A wizard's initial list of prepared spells seems to take no time at all, or at least it's assumed any time required has been spent before play begins:

You prepare the list of wizard spells that are available for you to cast. To do so, choose a number of wizard spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell).

The preparation rules are mostly the same for the other spell-preparing classes - Clerics, Druids and Paladins - with the same time taken to prepare "a new list" of spells.

I couldn't find any reference to any of these classes having to spend time on spell preparation if they're not changing their spells around, but equally it's not stated anywhere how long spells stay prepared. The general spellcasting rules have a section on "Known and Prepared Spells", but all it says on the matter is:

This process [of preparing spells] varies for different classes, as detailed in their descriptions.

The only hint I could find was in the Your Spellbook sidebar for wizards:

If you lose your spellbook, you can use the same procedure to transcribe the spells that you have prepared into a new spellbook.

For high-level wizards this could take multiple days, since this takes an hour per level per spell. But there's no reference here to a time limit or prepared spells becoming "unfixed" in the wizard's mind if they cannot access their spellbook.

My questions are:

• Once a wizard, cleric, druid or paladin has prepared a list of spells, do they stay prepared indefinitely?
• Does such a spellcaster only have to spend time preparing spells when changing that list?
• Does this mean a wizard only needs their spellbook to access spells they do not currently have prepared?

Answer 1

Short: Yes, Yes, Yes

The section for wizards prepared spells states --You can change your list of prepared spells when you finish a long rest-- This means you do not have to and can keep the old prepared spells. Also remember that casting a spell does not remove it from your list of prepared spells.

As per wizard Spellcasting feature

Casting the spell doesn't remove it from your list of prepared spells

If you are not preparing new spells you do not have to spend time on doing so.

And the wizards spell book indeed is mostly needed to prepare new spells and you can cast new spells perfectly well without it at hand. Though I would keep it near in case you need to prepare new spells and losing it would cost you a lot of time to write everything down again.