Theorem fprodcl

Description: Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017)

	Ref	Expression
Hypotheses	fprodcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
	fprodcl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in ℂ$$
Assertion	fprodcl	$$⊢\phi \to \prod _{k\in A}B\in ℂ$$

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fprodcl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in ℂ$$
3		ssidd	$$⊢\phi \to ℂ\subseteq ℂ$$
4		mulcl	$$⊢\left(x\in ℂ\wedge y\in ℂ\right)\to xy\in ℂ$$
5	4	adantl	$$⊢\left(\phi \wedge \left(x\in ℂ\wedge y\in ℂ\right)\right)\to xy\in ℂ$$
6		1cnd	$$⊢\phi \to 1\in ℂ$$
7	3 5 1 2 6	fprodcllem	$$⊢\phi \to \prod _{k\in A}B\in ℂ$$