Only for experts capable of following the proof. For everyone else, it's a magical formula.

Not to mention, I very very much doubt that those articles prove every property they present (since I've never seen a math text of any kind do that, essentially).

You’ll always need some axioms to build on.

If you allow that, proofs made using proof assistants prove every property they present.

And of course, Principia Mathematica (https://en.wikipedia.org/wiki/Principia_Mathematica) is a non-computer example.

Both get tedious fast if you try to understand a proof from the axioms up. That’s why you’ll rarely find a publication starting from almost zero.

Because that's essentially original research. If it's a known proof, it should exist in written form somewhere else, which you can cite. if it's novel, it belongs somewhere other than Wikipedia.

Because wikipedia is not trying to report the capital-T Truth. Its goal is to summarize human knowledge found by others.

You could cite all of the notations that you introduce because that tends to be a mess.

Aside from the fact that proofs are often difficult to verify, one cannot prove what a monoid or a vector space is. They are definitions, and to make sure the ones in the article match what is used in the mathematical community, you need citations.

And yes, sometimes there are conflicts. In France, we have two competing definition of a limit (relating to wether you include the point in its neighbourhood), one being the traditional one, taught in schools, and one being the one that's become the world standard and used from university onwards.

How do you arbiter that on wikipedia without sources ?

I always chuckle when I see some like basic logical steps slapped with "citation needed".

I similarly chuckle in cases where there's a citation covering the whole sentence but someone felt the need to slap a "citation needed" on a specific clause despite that clause being supported in the existing citation.