# Metamath Proof Explorer

## Theorem fprodcllem

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

Ref Expression
Hypotheses fprodcllem.1 ( 𝜑𝑆 ⊆ ℂ )
fprodcllem.2 ( ( 𝜑 ∧ ( 𝑥𝑆𝑦𝑆 ) ) → ( 𝑥 · 𝑦 ) ∈ 𝑆 )
fprodcllem.3 ( 𝜑𝐴 ∈ Fin )
fprodcllem.4 ( ( 𝜑𝑘𝐴 ) → 𝐵𝑆 )
fprodcllem.5 ( 𝜑 → 1 ∈ 𝑆 )
Assertion fprodcllem ( 𝜑 → ∏ 𝑘𝐴 𝐵𝑆 )

### Proof

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