This is corollary 5.39 in John Lee’s Introduction to Smooth Manifolds.

Corollary 5.39Suppose $ S \subseteq M$ is a level set of a smooth submersion $ \Phi=(\Phi^1,…\Phi^k): M \to \mathbb{R^k}$ . A vector $ v \in T_pM$ is tangent to $ S$ if and only if $ v\Phi^1=…=v\Phi^k=0$ .

He said that the proof is immediate. But I can’t figure it out.The author said it’s a restatement of Proposition 5.38 in a special case in which the defining function takes its value in $ \mathbb{R^k}$ .

Proposition 5.38Suppose $ M$ is a smooth manifold and $ S \subseteq M$ s an embedded submanifold. If $ \Phi:U \to N$ is any local defining function map for S, then $ T_pS = \mathrm{Ker}d\Phi_p:T_pM \to T_{\Phi(p)}N$ for each $ p \in S\cap U$ .

The step making me stuck is the sufficiency part. Suppose that $ v\Phi^1=…=v\Phi^k=0$ , in order to use Proposition 5.38, I want to show that $ v \in \mathrm{Ker}d\Phi_p$ , but I don’t know how to reach this…The author said this proof is immediate, so may I miss something? Thanks for your help in advance.