Let $X_1,...,X_n$ be a random sample from a Poisson distribution with mean $\lambda$ and $T = \sum_{i=1}^n X_i $ . Show that the distribution of $X_1,...,X_n$ is independant of $\lambda$ so that $T$ is a sufficient statistic for $\lambda$.

By definition of sufficient statistic : $$ P(X_1 = x_1,...,X_n=x_n | T =t ) = \frac{P(X_1=x1,...,X_n=x_n,T=t)}{P(T=t)}$$ According to the teaching assistant, this is equal to : $$ \frac{P(X_1=x1)...P(X_n=t-\sum_{i=1}^{n-1}x_i)}{P(T=t)}$$

I understand that because $X_1,...X_n$ are iid, probability of their intersection is product of their probabilities. I also understand that $ x_n=t-\sum_{i=1}^{n-1}x_i) $. What I cannot figure out is why the factor $P(T=t)$ disappears of the numerator.