Description: An infinite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 28-Apr-2014)
Ref | Expression | ||
---|---|---|---|
Hypotheses | isumrecl.1 | |
|
isumrecl.2 | |
||
isumrecl.3 | |
||
isumrecl.4 | |
||
isumrecl.5 | |
||
isumge0.6 | |
||
Assertion | isumge0 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumrecl.1 | |
|
2 | isumrecl.2 | |
|
3 | isumrecl.3 | |
|
4 | isumrecl.4 | |
|
5 | isumrecl.5 | |
|
6 | isumge0.6 | |
|
7 | 4 | recnd | |
8 | 1 2 3 7 5 | isumclim2 | |
9 | fveq2 | |
|
10 | 9 | cbvsumv | |
11 | 3 | sumeq2dv | |
12 | 10 11 | eqtrid | |
13 | 8 12 | breqtrrd | |
14 | 3 4 | eqeltrd | |
15 | 6 3 | breqtrrd | |
16 | 1 2 13 14 15 | iserge0 | |
17 | 16 12 | breqtrd | |