Theorem fsumrpcl

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

	Ref	Expression
Hypotheses	fsumcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
	fsumrpcl.2	$$⊢\phi \to A\ne \varnothing$$
	fsumrpcl.3	$$⊢\left(\phi \wedge k\in A\right)\to B\in {ℝ}^{+}$$
Assertion	fsumrpcl	$$⊢\phi \to \sum _{k\in A}B\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fsumrpcl.2	$$⊢\phi \to A\ne \varnothing$$
3		fsumrpcl.3	$$⊢\left(\phi \wedge k\in A\right)\to B\in {ℝ}^{+}$$
4		rpssre	$$⊢{ℝ}^{+}\subseteq ℝ$$
5		ax-resscn	$$⊢ℝ\subseteq ℂ$$
6	4 5	sstri	$$⊢{ℝ}^{+}\subseteq ℂ$$
7	6	a1i	$$⊢\phi \to {ℝ}^{+}\subseteq ℂ$$
8		rpaddcl	$$⊢\left(x\in {ℝ}^{+}\wedge y\in {ℝ}^{+}\right)\to x+y\in {ℝ}^{+}$$
9	8	adantl	$$⊢\left(\phi \wedge \left(x\in {ℝ}^{+}\wedge y\in {ℝ}^{+}\right)\right)\to x+y\in {ℝ}^{+}$$
10	7 9 1 3 2	fsumcl2lem	$$⊢\phi \to \sum _{k\in A}B\in {ℝ}^{+}$$