Product of Products

Theorem

Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers.

Let $\ds \prod_{R \paren i} x_i$ denote a continued product over $R$.

Then:

$\ds \prod_{R \paren i} \paren {b_i c_i} = \paren {\prod_{R \paren i} b_i} \paren {\prod_{R \paren i} c_i}$

Let $b_i =: a_{i 1}$ and $c_i =: a_{i 2}$.

Then:

$\ds \prod_{R \paren i} \paren {b_i c_i}$

$=$

$\ds \prod_{R \paren i} \paren {a_{i 1} a_{i 2} }$

by definition

$\ds $

$=$

$\ds \prod_{R \paren i} \paren {\prod_{1 \mathop \le j \mathop \le 2} a_{i j} }$

$\ds $

$=$

$\ds \prod_{1 \mathop \le j \mathop \le 2} \paren {\prod_{R \paren i} a_{i j} }$

$\ds $

$=$

$\ds \paren {\prod_{R \paren i} a_{i 1} } \paren {\prod_{R \paren i} a_{i 2} }$

$\ds $

$=$

$\ds \paren {\prod_{R \paren i} b_i} \paren {\prod_{R \paren i} c_i}$

by definition

$\blacksquare$