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Metamath Proof Explorer

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 )