Theorem fsumnn0cl

Description: Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015)

	Ref	Expression
Hypotheses	fsumcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumnn0cl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℕ0 )
Assertion	fsumnn0cl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsumnn0cl.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℕ0 )
3		nn0sscn	⊢ ℕ0 ⊆ ℂ
4	3	a1i	⊢ ( 𝜑 → ℕ0 ⊆ ℂ )
5		nn0addcl	⊢ ( ( 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) → ( 𝑥 + 𝑦 ) ∈ ℕ0 )
6	5	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) ) → ( 𝑥 + 𝑦 ) ∈ ℕ0 )
7		0nn0	⊢ 0 ∈ ℕ0
8	7	a1i	⊢ ( 𝜑 → 0 ∈ ℕ0 )
9	4 6 1 2 8	fsumcllem	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℕ0 )