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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 → 𝐴 ∈ Fin )
	fprodclf.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
Assertion	fprodclf	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		fprodclf.kph	⊢ Ⅎ 𝑘 𝜑
2		fprodclf.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
3		fprodclf.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
4		ssidd	⊢ ( 𝜑 → ℂ ⊆ ℂ )
5		mulcl	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 · 𝑦 ) ∈ ℂ )
6	5	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 · 𝑦 ) ∈ ℂ )
7		1cnd	⊢ ( 𝜑 → 1 ∈ ℂ )
8	1 4 6 2 3 7	fprodcllemf	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ ℂ )