Description: The arbitrary 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, 17-Aug-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | sge0fsummpt.a | |
|
sge0fsummpt.b | |
||
Assertion | sge0fsummpt | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0fsummpt.a | |
|
2 | sge0fsummpt.b | |
|
3 | eqid | |
|
4 | 2 3 | fmptd | |
5 | 1 4 | sge0fsum | |
6 | fveq2 | |
|
7 | nfcv | |
|
8 | nfcv | |
|
9 | nfmpt1 | |
|
10 | nfcv | |
|
11 | 9 10 | nffv | |
12 | nfcv | |
|
13 | 6 7 8 11 12 | cbvsum | |
14 | 13 | a1i | |
15 | simpr | |
|
16 | 3 | fvmpt2 | |
17 | 15 2 16 | syl2anc | |
18 | 17 | sumeq2dv | |
19 | 5 14 18 | 3eqtrd | |