In the book Algebraic Geometry II by Mumford and Oda (its draft here), I came across the following proposition in the section on smoothness:
Proposition 5.3.10 (page 171 in draft). If $f\colon X\to Y$ is a smooth morphism (of schemes) at $x\in X$ of relative dimension $n$ and $y=f(x)$, then $\operatorname{gr}\mathcal O_x$ is a polynomial ring with $n$ variables over $\operatorname{gr}(\mathcal O_y)\otimes_{\mathbb k(y)}\mathbb k(x)$ — more precisely, $\exists t_1,\dotsc,t_n\in\mathfrak m_x/\mathfrak m_x^2$ such that \begin{equation} \mathfrak m_x^\nu/\mathfrak m_x^{\nu+1}\cong \bigoplus_{l=0}^{\nu}\bigoplus_{\substack{\text{multi-indices}\\ \alpha,\, \lvert\alpha\rvert=\nu-l}}\bigl(\mathfrak m_y^l/\mathfrak m_y^{l+1}\otimes_{\mathbb k(y)}\mathbb k(x)\bigr)\cdot t^\alpha. \end{equation} Thus \begin{equation} \mathbf{TC}_{X,x}\cong\mathbf{TC}_{Y,y}\times_{\operatorname{Spec}\mathbb k(y)}\mathbb A^n_{\mathbb k(x)}. \end{equation} (Here $\mathbf{TC}$ denotes the tangent cone.)
After that, a corollary:
Corollary 5.3.15. If a $K$-variety $X$ is smooth of relative dimension $n$ over $K$ at some point $x\in X$, then $\dim X=n$.
Proof. Apply Proposition 5.3.10 to the generic point $\eta\in X$.
I do not understand how the proposition implies the corollary. Applying the proposition to the generic point $\eta$, it follows that $\operatorname{gr}\mathcal O_\eta=\operatorname{gr}\mathbf R(X)=\mathbf R(X)$, the function field, is a polynomial ring with $n$ variables over $\operatorname{gr}(K)\otimes_K\mathbf R(X)=\mathbf R(X)$. Then since $\mathbf R(X)$ is a field, $n$ must be $0$. What is wrong with my argument?
Any help would be appreciated. Thanks in advance.