Classical global estimate for $H^1$ error

I’m having lots of troubles in understanding the proof the estimation of the classical $H^1$ error using finite elements of degree $r$.

$$||u-u_h||_{H^1(Omega)} leq frac{M}{alpha} C h^r |u|_{H^{r+1}(Omega)}$$
where $M$ and $alpha$ are the operator norm and the coercivity constant, respectively and $C$ is a constand independent on $h$.

My book (Quarteroni – Numerical methods for differential problems) shows this by using the Cèa’s lemma and the estimate on the seminorm of the interpolation error. More precisely, it states:

$$||u-u_h||_{H^1(Omega)} leq frac{M}{alpha} inf_{v_h in V_h} ||u – v_h||_{H^1(Omega)} leq frac{M}{alpha} ||u-Pi^ru||_{H^1(Omega)}$$
Now use the estimate $$|u-Pi^r u|_{H^m(Omega)} leq Ch^{r+1-m} |u|_{H^{m}(Omega)}$$ (valid for $m=0,1$ and $rgeq 1$.
) with $m=1$ to conclude

The very last step is what I cannot do: I can’t see how to pass from the $H^1$ seminorm to the full $H^1$ norm. I mean, if I need to bound $||u-Pi^ru||_{H^1(Omega)}$, how can I use the bound on the seminorm?