Let k be a field, and let A be a local, noetherian, complete k-algebra with residue field k. Suppose that there are elements t_1,....t_n in the maximal ideal of A such that the map k[[X_1,....X_n]]---->A that sends X_i to t_i for all i is injective. Is the dimension of A greater or equal than n ?

Let $k$ be a field, and let $A$ be a local, noetherian, complete k-algebra with residue field $k$. Suppose that there are elements $t_1,\dots,t_n$ in the maximal ideal of $A$ such that the map $k[[X_1,\dots,X_n]] \rightarrow A$ that sends $X_i$ to $t_i$ for all $i$ is injective. Is the dimension of $A$ greater or equal than $n$ ?