Step |
Hyp |
Ref |
Expression |
1 |
|
sgnval |
|- ( A e. RR* -> ( sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) |
2 |
1
|
adantr |
|- ( ( A e. RR* /\ A < 0 ) -> ( sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) ) |
3 |
|
0xr |
|- 0 e. RR* |
4 |
|
xrltne |
|- ( ( A e. RR* /\ 0 e. RR* /\ A < 0 ) -> 0 =/= A ) |
5 |
3 4
|
mp3an2 |
|- ( ( A e. RR* /\ A < 0 ) -> 0 =/= A ) |
6 |
|
nesym |
|- ( 0 =/= A <-> -. A = 0 ) |
7 |
5 6
|
sylib |
|- ( ( A e. RR* /\ A < 0 ) -> -. A = 0 ) |
8 |
7
|
iffalsed |
|- ( ( A e. RR* /\ A < 0 ) -> if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) = if ( A < 0 , -u 1 , 1 ) ) |
9 |
|
iftrue |
|- ( A < 0 -> if ( A < 0 , -u 1 , 1 ) = -u 1 ) |
10 |
9
|
adantl |
|- ( ( A e. RR* /\ A < 0 ) -> if ( A < 0 , -u 1 , 1 ) = -u 1 ) |
11 |
2 8 10
|
3eqtrd |
|- ( ( A e. RR* /\ A < 0 ) -> ( sgn ` A ) = -u 1 ) |