Theorem sge0clmpt

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

	Ref	Expression
Hypotheses	sge0clmpt.xph	|- F/ x ph
	sge0clmpt.a	|- ( ph -> A e. V )
	sge0clmpt.b	|- ( ( ph /\ x e. A ) -> B e. ( 0 [,] +oo ) )
Assertion	sge0clmpt	|- ( ph -> ( sum^ ` ( x e. A |-> B ) ) e. ( 0 [,] +oo ) )

Proof

Step	Hyp	Ref	Expression
1		sge0clmpt.xph	|- F/ x ph
2		sge0clmpt.a	|- ( ph -> A e. V )
3		sge0clmpt.b	|- ( ( ph /\ x e. A ) -> B e. ( 0 [,] +oo ) )
4		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
5	1 3 4	fmptdf	|- ( ph -> ( x e. A |-> B ) : A --> ( 0 [,] +oo ) )
6	2 5	sge0cl	|- ( ph -> ( sum^ ` ( x e. A |-> B ) ) e. ( 0 [,] +oo ) )