Theorem fsumcl

Description: Closure of a finite sum of complex numbers A ( k ) . (Contributed by NM, 9-Nov-2005) (Revised by Mario Carneiro, 22-Apr-2014)

	Ref	Expression
Hypotheses	fsumcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumcl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
Assertion	fsumcl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsumcl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
3		ssidd	⊢ ( 𝜑 → ℂ ⊆ ℂ )
4		addcl	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + 𝑦 ) ∈ ℂ )
5	4	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 + 𝑦 ) ∈ ℂ )
6		0cnd	⊢ ( 𝜑 → 0 ∈ ℂ )
7	3 5 1 2 6	fsumcllem	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℂ )