Lemma 35.29.1. The property $\mathcal{P}(f)=$“$f$ is smooth” is smooth local on the source.

**Proof.**
Combine Lemma 35.25.4 with Morphisms, Lemma 29.34.2 (local for Zariski on source and target), Morphisms, Lemma 29.34.4 (pre-composing), and Lemma 35.13.4 (part (4)).
$\square$

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