Description: Any simple function takes the value zero on a set of unbounded measure, so in particular this set is not empty. (Contributed by Mario Carneiro, 18-Jun-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | i1f0rn | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnre | |
|
2 | 1 | neli | |
3 | rembl | |
|
4 | mblvol | |
|
5 | 3 4 | ax-mp | |
6 | ovolre | |
|
7 | 5 6 | eqtri | |
8 | cnvimarndm | |
|
9 | i1ff | |
|
10 | 9 | fdmd | |
11 | 10 | adantr | |
12 | 8 11 | eqtrid | |
13 | 12 | fveq2d | |
14 | i1fima2 | |
|
15 | 13 14 | eqeltrrd | |
16 | 7 15 | eqeltrrid | |
17 | 16 | ex | |
18 | 2 17 | mt3i | |