Theorem sge0xrclmpt

Description: The generalized sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020)

	Ref	Expression
Hypotheses	sge0xrclmpt.xph	|- F/ x ph
	sge0xrclmpt.a	|- ( ph -> A e. V )
	sge0xrclmpt.b	|- ( ( ph /\ x e. A ) -> B e. ( 0 [,] +oo ) )
Assertion	sge0xrclmpt	|- ( ph -> ( sum^ ` ( x e. A |-> B ) ) e. RR* )

Proof

Step	Hyp	Ref	Expression
1		sge0xrclmpt.xph	|- F/ x ph
2		sge0xrclmpt.a	|- ( ph -> A e. V )
3		sge0xrclmpt.b	|- ( ( ph /\ x e. A ) -> B e. ( 0 [,] +oo ) )
4		iccssxr	|- ( 0 [,] +oo ) C_ RR*
5	1 2 3	sge0clmpt	|- ( ph -> ( sum^ ` ( x e. A |-> B ) ) e. ( 0 [,] +oo ) )
6	4 5	sseldi	|- ( ph -> ( sum^ ` ( x e. A |-> B ) ) e. RR* )