Theorem fprodcllem

Description: Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017)

	Ref	Expression
Hypotheses	fprodcllem.1	⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
	fprodcllem.2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 · 𝑦 ) ∈ 𝑆 )
	fprodcllem.3	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fprodcllem.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑆 )
	fprodcllem.5	⊢ ( 𝜑 → 1 ∈ 𝑆 )
Assertion	fprodcllem	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		fprodcllem.1	⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
2		fprodcllem.2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 · 𝑦 ) ∈ 𝑆 )
3		fprodcllem.3	⊢ ( 𝜑 → 𝐴 ∈ Fin )
4		fprodcllem.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑆 )
5		fprodcllem.5	⊢ ( 𝜑 → 1 ∈ 𝑆 )
6		prodeq1	⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ ∅ 𝐵 )
7		prod0	⊢ ∏ 𝑘 ∈ ∅ 𝐵 = 1
8	6 7	eqtrdi	⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐵 = 1 )
9	8	adantl	⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → ∏ 𝑘 ∈ 𝐴 𝐵 = 1 )
10	5	adantr	⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → 1 ∈ 𝑆 )
11	9 10	eqeltrd	⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )
12	1	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → 𝑆 ⊆ ℂ )
13	2	adantlr	⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 · 𝑦 ) ∈ 𝑆 )
14	3	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → 𝐴 ∈ Fin )
15	4	adantlr	⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑆 )
16		simpr	⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → 𝐴 ≠ ∅ )
17	12 13 14 15 16	fprodcl2lem	⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )
18	11 17	pm2.61dane	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )