Product of Products

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Theorem

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

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


Let the fiber of truth of $R$ be finite.

Then:

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


Proof

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

Then:

\(\displaystyle \prod_{R \paren i} \paren {b_i c_i}\) \(=\) \(\displaystyle \prod_{R \paren i} \paren {a_{i 1} a_{i 2} }\) by definition
\(\displaystyle \) \(=\) \(\displaystyle \prod_{R \paren i} \paren {\prod_{1 \mathop \le j \mathop \le 2} a_{i j} }\) Definition of Product by Propositional Function
\(\displaystyle \) \(=\) \(\displaystyle \prod_{1 \mathop \le j \mathop \le 2} \paren {\prod_{R \paren i} a_{i j} }\) Exchange of Order of Product
\(\displaystyle \) \(=\) \(\displaystyle \paren {\prod_{R \paren i} a_{i 1} } \paren {\prod_{R \paren i} a_{i 2} }\) Definition of Product by Propositional Function
\(\displaystyle \) \(=\) \(\displaystyle \paren {\prod_{R \paren i} b_i} \paren {\prod_{R \paren i} c_i}\) by definition

$\blacksquare$


Sources