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Metamath Proof Explorer

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 )