I'm reading a proof about Langrangian => Hamiltonian and one part of it just doesn't make sense to me.

The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the Hamiltonian is defined via the Legendre transform $$H(p,q,t) = \max_{\dot q} [p \cdot \dot q - L(q, \dot q, t)]$$

Under the right conditions there exists a function $\dot Q (p,q,t)$ such that $$H(p,q,t) = p \cdot \dot Q(p,q,t) - L(q, \dot Q(p,q,t), t)$$ i.e. when $\dot Q(p,q,t)$ satisfies $p = \frac{\partial L}{\partial \dot q} (q, \dot Q(p,q,t), t)$ (Finding this function is usually called "inverting p")

By taking partials in the $p$ variable and using the relationship, we can obtain the relationship $$\dot Q = \frac{\partial H}{\partial p}$$

Because of the notation I chose, I get the strong urge to say $\dot q = \frac{\partial H}{\partial p}$ , and in fact this is what the textbook does. But have we proved this?

In other words, how can we deduce that $$q'(t) = \frac{\partial H}{\partial p}(p(t), q'(t), t)$$ for any differentiable vector valued function $q$? (or maybe there are more conditions we need on $q$? Here $$p(t) = \frac{\partial L}{\partial \dot q}(q(t), q'(t), t)$$ according to Lagrange's equations.