Theorem sge0ssrempt

Description: If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	sge0ssrempt.xph	|- F/ x ph
	sge0ssrempt.a	|- ( ph -> A e. V )
	sge0ssrempt.b	|- ( ( ph /\ x e. A ) -> B e. ( 0 [,] +oo ) )
	sge0ssrempt.re	|- ( ph -> ( sum^ ` ( x e. A |-> B ) ) e. RR )
	sge0ssrempt.c	|- ( ph -> C C_ A )
Assertion	sge0ssrempt	|- ( ph -> ( sum^ ` ( x e. C |-> B ) ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		sge0ssrempt.xph	|- F/ x ph
2		sge0ssrempt.a	|- ( ph -> A e. V )
3		sge0ssrempt.b	|- ( ( ph /\ x e. A ) -> B e. ( 0 [,] +oo ) )
4		sge0ssrempt.re	|- ( ph -> ( sum^ ` ( x e. A |-> B ) ) e. RR )
5		sge0ssrempt.c	|- ( ph -> C C_ A )
6	5	resmptd	|- ( ph -> ( ( x e. A |-> B ) |` C ) = ( x e. C |-> B ) )
7	6	fveq2d	|- ( ph -> ( sum^ ` ( ( x e. A |-> B ) |` C ) ) = ( sum^ ` ( x e. C |-> B ) ) )
8	7	eqcomd	|- ( ph -> ( sum^ ` ( x e. C |-> B ) ) = ( sum^ ` ( ( x e. A |-> B ) |` C ) ) )
9		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
10	1 3 9	fmptdf	|- ( ph -> ( x e. A |-> B ) : A --> ( 0 [,] +oo ) )
11	2 10 4	sge0ssre	|- ( ph -> ( sum^ ` ( ( x e. A |-> B ) |` C ) ) e. RR )
12	8 11	eqeltrd	|- ( ph -> ( sum^ ` ( x e. C |-> B ) ) e. RR )