Theorem fsumrpcl

Description: Closure of a finite sum of positive reals. (Contributed by Mario Carneiro, 3-Jun-2014)

	Ref	Expression
Hypotheses	fsumcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumrpcl.2	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
	fsumrpcl.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ+ )
Assertion	fsumrpcl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsumrpcl.2	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
3		fsumrpcl.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ+ )
4		rpssre	⊢ ℝ+ ⊆ ℝ
5		ax-resscn	⊢ ℝ ⊆ ℂ
6	4 5	sstri	⊢ ℝ+ ⊆ ℂ
7	6	a1i	⊢ ( 𝜑 → ℝ+ ⊆ ℂ )
8		rpaddcl	⊢ ( ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) → ( 𝑥 + 𝑦 ) ∈ ℝ+ )
9	8	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) ) → ( 𝑥 + 𝑦 ) ∈ ℝ+ )
10	7 9 1 3 2	fsumcl2lem	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℝ+ )