How exactly are we proving that modus ponens is valid?

Question

This is being done in the context of a Hilbert system in propositional logic.

But let's assume we've already proven all the axioms are tautologies / will always be true.

How then are we showing that modus ponens is also valid? (or is valid the wrong word to use here? Normally, "valid" for a statement means the logic holds IF the premises were true, but premises being true is what makes it actually sound... but when it comes to inference rules I don't know the terminology)

$P, (P \to Q) \vdash Q$

Is this strictly an informal "look at the truth tables and reason it out" kind of thing?

i.e. if we assume $P$ is true and $P \to Q$ is true because they are each reducible to either true axioms or true atomic propositions, then we know $P \to Q = \top$ becomes $\top \to Q = \top$. And in the truth tables for $\to$, this only has one solution, i.e. where $Q = \top$, therefore modus ponens holds? Or are we now showing that $P, (P \to Q) \vDash Q$?

I mean is that it? Or is it a more formal approach? I get totally lost when it comes to understanding and manipulating / proving things with these metalogical symbols.

We prove the soundness of the inference rules (like e.g. modus ponens). See the post Method of Proving Soundness of Propositional Logic as well as your previous post Soundness vs completeness, am I understanding ? And proving soundness ? — Mauro ALLEGRANZA, Sep 10 '18 at 11:31
The Double turnstile symbol has two uses (strictly linked) in logic. $\vDash \varphi$ means that formula $\varphi$ is valid (true in every interpretation). $\Gamma \vDash \varphi$ means that formula $\varphi$ is logical consequence of the set $\Gamma$ of formulas. See your previous post How is my understanding of ⊢,⊨,→, etc. — Mauro ALLEGRANZA, Sep 10 '18 at 12:45
See also https://math.stackexchange.com/questions/281208/what-is-the-difference-between-a-sound-argument-and-a-valid-argument which seems to contradict this definition — user525966, Sep 10 '18 at 14:54
@MauroALLEGRANZA From your old post: "An argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required that a valid argument have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion." — user525966, Sep 10 '18 at 23:00

How exactly are we proving that modus ponens is valid?

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