Definition 3

Tag 0A7H, permanent. Cite it as [Bourbaki, Tag 0A7H].

Lie Groups and Lie Algebras, chapter III, § 1. Lie groups

Let G be a Lie group. A subset H of G is called a Lie subgroup if H is a subgroup and a submanifold of G.

An open subgroup of G is a Lie subgroup of G. In particular, if G is a real or complex Lie group, its identity component is a Lie subgroup of G.

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

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