orbit - Derivation of the formula for longitude of ascending node for a satellite

I've been looking into the document IS-GPS-200H to understand how to calculate satellite location in the ECEF coordinate.

I am having problem understanding the formula to derive $Omega$, the longitude of the ascending node (LAN) relative to Greenwich at given time $t$:

$$
Omega = Omega_0 + left( dot{Omega - w} right)times t_k - w times t_{oe}
$$

where:
$$
Omega_0: text{LAN relative to vernal equinox, at the beginning of the week}\
dot{Omega}: text{angular velocity for LAN, relative to vernal equinox.}\
w: text{angular velocity of earth, relative to vernal equinox.}\
t_k: t - t_{oe}\
t_{oe}: text{ephemeris reference epoch}\
$$
(and let us denote the beginning of the week as $t_0$ for brevity).

But if I try to work out this from scratch:

1. At $t = t_0$, LAN was $Omega_0$. But since what we really need is the difference of LAN and longitude of Greenwich, we also need to know $w_0$, the initial longitude of Greenwich at $t = t_0$.
   $$
   Omega(t = t_0) = Omega_0 - w_0
   $$


2. At the ephemeris reference epoch time $t = t_{oe}$, LAN and the earth both rotate with their respective angular momentum and hence:
   $$
   Omega(t = toe) = Omega_0 + w_0 + (dot{Omega} - w) times t_{oe}
   $$


3. As time varies from $toe$ to $t$, again LAN and the earth both rotate with their own respective angular momentum and hence
   $$
   Omega(t) = Omega_0 + w_0 + (dot{Omega} - w) times t_{oe} + (dot{Omega} - w) times t_k
   $$
   which obviously differs from the right formula by $w_0 + dot{Omega} times t_{oe}$.

My question is where am I making mistakes/misunderstanding the eqution?
Explain also why we don't need to know $w_0$ or equivalent input, that would be greatly appreciated.