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Metamath Proof Explorer

Theorem fprodclf

Description: Closure of a finite product of complex numbers. A version of fprodcl using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	fprodclf.kph	$⊢ Ⅎ k φ$
	fprodclf.a	$⊢ φ \to A \in Fin$
	fprodclf.b	$⊢ (φ \land k \in A) \to B \in ℂ$
Assertion	fprodclf	$⊢ φ \to \prod_{k \in A} B \in ℂ$

Proof

Step	Hyp	Ref	Expression
1		fprodclf.kph	$⊢ Ⅎ k φ$
2		fprodclf.a	$⊢ φ \to A \in Fin$
3		fprodclf.b	$⊢ (φ \land k \in A) \to B \in ℂ$
4		ssidd	$⊢ φ \to ℂ \subseteq ℂ$
5		mulcl	$⊢ (x \in ℂ \land y \in ℂ) \to x y \in ℂ$
6	5	adantl	$⊢ (φ \land (x \in ℂ \land y \in ℂ)) \to x y \in ℂ$
7		1cnd	$⊢ φ \to 1 \in ℂ$
8	1 4 6 2 3 7	fprodcllemf	$⊢ φ \to \prod_{k \in A} B \in ℂ$