Theorem fsumcllem

Description: - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005) (Revised by Mario Carneiro, 3-Jun-2014)

	Ref	Expression
Hypotheses	fsumcllem.1	⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
	fsumcllem.2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
	fsumcllem.3	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumcllem.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑆 )
	fsumcllem.5	⊢ ( 𝜑 → 0 ∈ 𝑆 )
Assertion	fsumcllem	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		fsumcllem.1	⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
2		fsumcllem.2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
3		fsumcllem.3	⊢ ( 𝜑 → 𝐴 ∈ Fin )
4		fsumcllem.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑆 )
5		fsumcllem.5	⊢ ( 𝜑 → 0 ∈ 𝑆 )
6		simpr	⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → 𝐴 = ∅ )
7	6	sumeq1d	⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ ∅ 𝐵 )
8		sum0	⊢ Σ 𝑘 ∈ ∅ 𝐵 = 0
9	7 8	eqtrdi	⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → Σ 𝑘 ∈ 𝐴 𝐵 = 0 )
10	5	adantr	⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → 0 ∈ 𝑆 )
11	9 10	eqeltrd	⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → Σ 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	fsumcl2lem	⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )
18	11 17	pm2.61dane	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )