Let $\Gamma_0 = \mathrm{Gal}(\mathbb{Q}_p(\zeta_{p^\infty})/\mathbb{Q_p})\cong \mathbb{Z}_p^*$, $\Gamma_0\supset\Gamma_n\cong1+p^n\mathbb{Z}_p$ for $n\geq 1$. Then we have $\Gamma_0=\Delta\times\Gamma_1$, where $\Delta=\mu_{p-1}$ Now we consider the Iwasawa algebra over $\Gamma_0$, I want to prove that $\mathbb{Z}_p[[\Gamma_0]]=\mathbb{Z}_p[\Delta]\otimes\mathbb{Z}_p[[\Gamma_1]]$. Here is my solution:
$$\mathbb{Z}_p[[\Gamma_0]]=\varprojlim_{m}\mathbb{Z}_p[\Gamma_0/\Gamma_m]=\varprojlim_{m}\mathbb{Z}_p[(\Delta\times\Gamma_1)/\Gamma_m]=\varprojlim_{m}\mathbb{Z}_p[\Delta][\Gamma_1/\Gamma_m]=\mathbb{Z}_p[\Delta]\otimes\mathbb{Z}_p[[\Gamma_1]] \text{.}$$
What makes me confused is the third equal sign, since before this equal sign the inverse limit runs over 0,1,2,..., while the limit runs over 1,2,3,... after this sign.
Will this change cause any problem?