Proposition 6
Tag 064D, permanent. Cite it as [Bourbaki, Tag 064D].
Algebra, chapter III, § 8. Determinants
For the columns of a square matrix X of finite order to be linearly independent, it is necessary and sufficient that det X be not a divisor of zero in A.
This follows from no. 2, Proposition 3.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.