Corollary 2

Tag 05H5, permanent. Cite it as [Bourbaki, Tag 05H5].

Algebra, chapter I, § 8. Rings

*Let A be a commutative ring, X a set, $\mathbf{u} = (u_x)_{x \in X}$ and $\mathbf{v} = (v_x)_{x \in X}$ two families of elements of A. Let $\mathbf{u} + \mathbf{v}$ denote the family $(u_x + v_x)_{x \in X}$ . For all $\lambda \in \mathbf{N}^{(X)}$ we write $\lambda! = \prod_{x \in X} \lambda(x)!$ . Then for all $\alpha \in \mathbf{N}^{(X)}$ , in the notation of § 7, no. 8,

(\mathbf{u} + \mathbf{v})^\alpha = \sum_{\beta + \gamma = \alpha} \frac{\alpha!}{\beta! \gamma!} \mathbf{u}^\beta \mathbf{v}^\gamma.

For $x \in X$ ,

(u_x + v_x)^{\alpha(x)} = \sum_{m+n=\alpha(x)} \frac{\alpha(x)!}{m! n!} u_x^m v_x^n

by Corollary 1. Taking the product of these equations for $x \in X$ and using (13), we obtain the corollary.

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

Cited by

Corollary, § 1
§ 4. Formal power series

Cites

§ 7, no. 8
Corollary 1