Theorem sge0repnfmpt

Description: The of nonnegative extended reals is a real number if and only if it is not +oo . (Contributed by Glauco Siliprandi, 21-Nov-2020)

	Ref	Expression
Hypotheses	sge0repnfmpt.k	|- F/ k ph
	sge0repnfmpt.a	|- ( ph -> A e. V )
	sge0repnfmpt.b	|- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )
Assertion	sge0repnfmpt	|- ( ph -> ( ( sum^ ` ( k e. A |-> B ) ) e. RR <-> -. ( sum^ ` ( k e. A |-> B ) ) = +oo ) )

Proof

Step	Hyp	Ref	Expression
1		sge0repnfmpt.k	|- F/ k ph
2		sge0repnfmpt.a	|- ( ph -> A e. V )
3		sge0repnfmpt.b	|- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )
4		eqid	|- ( k e. A |-> B ) = ( k e. A |-> B )
5	1 3 4	fmptdf	|- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) )
6	2 5	sge0repnf	|- ( ph -> ( ( sum^ ` ( k e. A |-> B ) ) e. RR <-> -. ( sum^ ` ( k e. A |-> B ) ) = +oo ) )