Proposition 1

Tag 0AH2, permanent. Cite it as [Bourbaki, Tag 0AH2].

Lie Groups and Lie Algebras, chapter III, § 7. Lie groups over an ultrametric field

Let G be a Lie group germ with identity element e. There exists a fundamental system of open neighbourhoods of e in G consisting of Lie subgroups of G.

Let L(G) be given a norm compatible with its topology and such that $\| [x, y] \| \leq \| x \| \| y \|$ for all x, y in L(G). Let G_1 be the Lie group defined by L(G). By § 4, no. 2, Theorem 2, G and G_1 are locally isomorphic. Then it suffices to apply § 4, no. 2, Lemma 3 (iii).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Theorem 2, § 10
Theorem 2, § 7
Theorem 3, § 7
Proposition 1, § 8

Cites

§ 4, no. 2, Lemma 3
§ 4, no. 2, Theorem 2