Theorem sge0isummpt

Description: If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020)

	Ref	Expression
Hypotheses	sge0isummpt.kph	|- F/ k ph
	sge0isummpt.a	|- ( ( ph /\ k e. Z ) -> A e. ( 0 [,) +oo ) )
	sge0isummpt.m	|- ( ph -> M e. ZZ )
	sge0isummpt.z	|- Z = ( ZZ>= ` M )
	sge0isummpt.b	|- ( ph -> seq M ( + , ( k e. Z |-> A ) ) ~~> B )
Assertion	sge0isummpt	|- ( ph -> ( sum^ ` ( k e. Z |-> A ) ) = B )

Proof

Step	Hyp	Ref	Expression
1		sge0isummpt.kph	|- F/ k ph
2		sge0isummpt.a	|- ( ( ph /\ k e. Z ) -> A e. ( 0 [,) +oo ) )
3		sge0isummpt.m	|- ( ph -> M e. ZZ )
4		sge0isummpt.z	|- Z = ( ZZ>= ` M )
5		sge0isummpt.b	|- ( ph -> seq M ( + , ( k e. Z |-> A ) ) ~~> B )
6		eqid	|- ( k e. Z |-> A ) = ( k e. Z |-> A )
7	1 2 6	fmptdf	|- ( ph -> ( k e. Z |-> A ) : Z --> ( 0 [,) +oo ) )
8		eqid	|- seq M ( + , ( k e. Z |-> A ) ) = seq M ( + , ( k e. Z |-> A ) )
9	3 4 7 8 5	sge0isum	|- ( ph -> ( sum^ ` ( k e. Z |-> A ) ) = B )