Description: Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | sgncl | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr | |
|
2 | 1 | fveq2d | |
3 | sgn0 | |
|
4 | 2 3 | eqtrdi | |
5 | c0ex | |
|
6 | 5 | tpid2 | |
7 | 4 6 | eqeltrdi | |
8 | sgnn | |
|
9 | negex | |
|
10 | 9 | tpid1 | |
11 | 8 10 | eqeltrdi | |
12 | 11 | adantlr | |
13 | sgnp | |
|
14 | 1ex | |
|
15 | 14 | tpid3 | |
16 | 13 15 | eqeltrdi | |
17 | 16 | adantlr | |
18 | 0xr | |
|
19 | xrlttri2 | |
|
20 | 19 | biimpa | |
21 | 18 20 | mpanl2 | |
22 | 12 17 21 | mpjaodan | |
23 | 7 22 | pm2.61dane | |