Theorem sge0ge0mpt

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

	Ref	Expression
Hypotheses	sge0ge0mpt.k	|- F/ k ph
	sge0ge0mpt.a	|- ( ph -> A e. V )
	sge0ge0mpt.b	|- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )
Assertion	sge0ge0mpt	|- ( ph -> 0 <_ ( sum^ ` ( k e. A |-> B ) ) )

Proof

Step	Hyp	Ref	Expression
1		sge0ge0mpt.k	|- F/ k ph
2		sge0ge0mpt.a	|- ( ph -> A e. V )
3		sge0ge0mpt.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	sge0ge0	|- ( ph -> 0 <_ ( sum^ ` ( k e. A |-> B ) ) )