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When I asked for proof of Ampere's circuital law I learned that this law itself is taken as an axiom. Okay I just need to know how did he come up with such a law? how to confirm that result experimentally? Is it just a total experimental law? When I searched a bit more I found a way to derive it from biot and savart law and I'm good with that but what I wanna know how to come up with it in the first place?

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Perhaps we should reviewwhat "proof" and "law" mean in Physics. A 'law' is, as you suggest, similar to a mathematical axiom. The difference is that physics-laws a selected to be compatible with our observable universe and that we don't generally find much use for investigating the results of choosing an alternative axiom (compare with nonEuclidean geometry, which was of purely theoretical use for a long time).

And for "proof," strictly speaking we are trying to show that all observable and testable behavior is consistent with the proposed axiom. Again, in mathematics, an axiom is just a starting point for constructing a system (think of it as a mathematical universe), but in physics there's not a lot of work <<\cough>>string theory<<\cough>> done with non-real-universe axioms.

For Ampere & Maxwell & Gauss, then, the 4 E-M equations were born of experiment, observation, and deep insight, after which a zillion experiments validated their utility.

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