Theorem fprodcllemf

Description: Finite product closure lemma. A version of fprodcllem using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	fprodcllemf.ph	⊢ Ⅎ 𝑘 𝜑
	fprodcllemf.s	⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
	fprodcllemf.xy	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 · 𝑦 ) ∈ 𝑆 )
	fprodcllemf.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fprodcllemf.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑆 )
	fprodcllemf.1	⊢ ( 𝜑 → 1 ∈ 𝑆 )
Assertion	fprodcllemf	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		fprodcllemf.ph	⊢ Ⅎ 𝑘 𝜑
2		fprodcllemf.s	⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
3		fprodcllemf.xy	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 · 𝑦 ) ∈ 𝑆 )
4		fprodcllemf.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
5		fprodcllemf.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑆 )
6		fprodcllemf.1	⊢ ( 𝜑 → 1 ∈ 𝑆 )
7		nfcv	⊢ Ⅎ 𝑗 𝐵
8		nfcsb1v	⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐵
9		csbeq1a	⊢ ( 𝑘 = 𝑗 → 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 )
10	7 8 9	cbvprodi	⊢ ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑗 ∈ 𝐴 ⦋ 𝑗 / 𝑘 ⦌ 𝐵
11	5	ex	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 → 𝐵 ∈ 𝑆 ) )
12	1 11	ralrimi	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )
13		rspsbc	⊢ ( 𝑗 ∈ 𝐴 → ( ∀ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → [ 𝑗 / 𝑘 ] 𝐵 ∈ 𝑆 ) )
14	12 13	mpan9	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → [ 𝑗 / 𝑘 ] 𝐵 ∈ 𝑆 )
15		sbcel1g	⊢ ( 𝑗 ∈ V → ( [ 𝑗 / 𝑘 ] 𝐵 ∈ 𝑆 ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑆 ) )
16	15	elv	⊢ ( [ 𝑗 / 𝑘 ] 𝐵 ∈ 𝑆 ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑆 )
17	14 16	sylib	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑆 )
18	2 3 4 17 6	fprodcllem	⊢ ( 𝜑 → ∏ 𝑗 ∈ 𝐴 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑆 )
19	10 18	eqeltrid	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )