How do I change the base year of real GDP using the GDP deflator and nominal GDP? I'll use the US as an example. I have three data series


nominal GDP $(Y)$
real GDP in 2005 USD $(\bar{Y})$
the GDP deflator $(d)$, with 2005 as the base year, so $d_{2005} = 100$


I want to change the base year to 2000. Are these calculations accurate? I use the notation $\$_{t}$ for USD in year $t$ prices to help myself keep the units straight.

My goal is $\bar{Y}_{t} \ \$_{2000}$.
\begin{align}
\frac{d_{2000}}{d_{t}} \cdot Y_{t} \ \$_t
&amp;= \frac{Y_{2000} \ \$_{2000}}{\bar{Y}_{2000} \ \$_{2005}} \cdot \frac{\bar{Y}_{t} \ \$_{2005}}{Y_{t} \ \$_{t}} \cdot Y_{t} \ \$_t \\
&amp;= \frac{Y_{2000} \ \$_{2000}}{\bar{Y}_{2000}} \cdot \bar{Y}_{t} \\
&amp;= \bar{Y}_{t} \ \$_{2000} \cdot \frac{Y_{2000}}{\bar{Y}_{2000}}
\end{align}

I think those are all the correct unit cancellations, but now I'm stuck with the unitless quantity $\frac{Y_{2000}}{\bar{Y}_{2000}}$, so I don't know how to complete the conversion. 

Am I doing this right?
It's just real GDP in year t, times (deflator in year 2000 / deflator in year t).