Metamath Proof Explorer


Theorem sge0fsummptf

Description: The generalized sum of a finite set of nonnegative extended real numbers is equal to the sum of those numbers, when none of them is +oo (Contributed by Glauco Siliprandi, 21-Nov-2020)

Ref Expression
Hypotheses sge0fsummptf.k k φ
sge0fsummptf.a φ A Fin
sge0fsummptf.b φ k A B 0 +∞
Assertion sge0fsummptf φ sum^ k A B = k A B

Proof

Step Hyp Ref Expression
1 sge0fsummptf.k k φ
2 sge0fsummptf.a φ A Fin
3 sge0fsummptf.b φ k A B 0 +∞
4 eqid k A B = k A B
5 1 3 4 fmptdf φ k A B : A 0 +∞
6 2 5 sge0fsum φ sum^ k A B = j A k A B j
7 fveq2 j = k k A B j = k A B k
8 nfcv _ k A
9 nfcv _ j A
10 nfmpt1 _ k k A B
11 nfcv _ k j
12 10 11 nffv _ k k A B j
13 nfcv _ j k A B k
14 7 8 9 12 13 cbvsum j A k A B j = k A k A B k
15 14 a1i φ j A k A B j = k A k A B k
16 simpr φ k A k A
17 4 fvmpt2 k A B 0 +∞ k A B k = B
18 16 3 17 syl2anc φ k A k A B k = B
19 18 ex φ k A k A B k = B
20 1 19 ralrimi φ k A k A B k = B
21 20 sumeq2d φ k A k A B k = k A B
22 6 15 21 3eqtrd φ sum^ k A B = k A B