Prove conservation law in quantum mechanics I major in Math, and I am studying Quantum Mechanics (QM). 
I see the conservation law in QM as a mathematical theorem.
Please check if my understanding is right, and help me to prove the theorem?
Background (might be wrong): 
Quantum state is described by wave function $\Psi (r,t)$, and follows Schrödinger equation: 
$$H\Psi (r,t)=i\hbar \frac{\partial \Psi(r,t)}{\partial t} \tag{1}$$
An observable $O$ is conserved if there are quantum states $\Psi(r,t)$ satisfying: 
$$O\Psi(r,t) = \lambda \Psi(r,t) \quad\text{ for some } \lambda \neq 0 \text{ and every } t \tag{2}$$
Theorem: An observable is conserved if its associated Hermitian operator $O$ commutes with Hamiltonian operator, i.e.: $$[H, O] = 0 \tag{3}.$$
Question: How to prove (3) from (1) and (2)?
 A: (2) says that if $\psi$ is an eigenvector of $\hat{O}$ at some time $t=0$, it stays at eigenvector of $\hat{O}$ thereafter. For that to be true, $\psi$ must also be an eigenvector of $\hat{H}$ since, infinitesimally, 
$$\psi(t+\delta t) = \psi(t) + \delta t \frac{\partial}{\partial t} \psi(t).$$
From (1), you know that this means
$$\psi(t+\delta t) = \psi(t) + (\delta t) \frac{\hat{H}}{i \hbar} \psi(t) $$
and so $\hat{H} \psi = E \psi$. Now you have proven there are simultaneous eigenvectors of $\hat{H}$ and $\hat{O}$, which is the same as $[\hat{H},\hat{O}] = 0$
A: The most elegant way under my perspective is try to transform (1) to an analogous equation  given by the Heisenberg equation, but this can lead to confusion.
In order to keep it simple, let suppose that $H$ and $O$ are not time dependent (in case they were we can, in principle move the time dependence to the states instead of the operators, c.f. Schrödinger picture), therefore
$$
\partial_t O = 0\quad \partial_t H = 0,
$$
where $\partial_t\cdot = \frac{\partial\cdot}{\partial t}$. Assume $O$ is an observable, and let $\psi(t,\vec{r})\equiv\psi$ be an eigenstate of $O$, i.e. $O\psi = \lambda\psi$. Then:
$$
[H,O]\psi = HO\psi-OH\psi =\\= H(O\psi)-O(H\psi) =\\= H\lambda\psi-Oi \hbar\partial_t\psi  =\\= \lambda H\psi-i \hbar\partial_t O\psi =\\= \lambda i\hbar\partial_t\psi-i \hbar\partial_t \lambda\psi =\\=
 i\hbar\lambda \partial_t\psi-i \hbar\lambda\partial_t \psi =0\quad \blacksquare
$$
But as stated in the comments, there are several issues in this concept: this relation holds only under the statements considered and, although is related to the concept of conserved quantities in QM, must be taken carefully.
