Theorem sge0xrcl

Description: The arbitrary sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	sge0xrcl.x	|- ( ph -> X e. V )
	sge0xrcl.f	|- ( ph -> F : X --> ( 0 [,] +oo ) )
Assertion	sge0xrcl	|- ( ph -> ( sum^ ` F ) e. RR* )

Proof

Step	Hyp	Ref	Expression
1		sge0xrcl.x	|- ( ph -> X e. V )
2		sge0xrcl.f	|- ( ph -> F : X --> ( 0 [,] +oo ) )
3		iccssxr	|- ( 0 [,] +oo ) C_ RR*
4	1 2	sge0cl	|- ( ph -> ( sum^ ` F ) e. ( 0 [,] +oo ) )
5	3 4	sseldi	|- ( ph -> ( sum^ ` F ) e. RR* )