Step |
Hyp |
Ref |
Expression |
1 |
|
ine0 |
|- _i =/= 0 |
2 |
1
|
neii |
|- -. _i = 0 |
3 |
|
0lt1 |
|- 0 < 1 |
4 |
|
0re |
|- 0 e. RR |
5 |
|
1re |
|- 1 e. RR |
6 |
4 5
|
ltnsymi |
|- ( 0 < 1 -> -. 1 < 0 ) |
7 |
3 6
|
ax-mp |
|- -. 1 < 0 |
8 |
|
ixi |
|- ( _i x. _i ) = -u 1 |
9 |
5
|
renegcli |
|- -u 1 e. RR |
10 |
8 9
|
eqeltri |
|- ( _i x. _i ) e. RR |
11 |
4 10 5
|
ltadd1i |
|- ( 0 < ( _i x. _i ) <-> ( 0 + 1 ) < ( ( _i x. _i ) + 1 ) ) |
12 |
|
ax-1cn |
|- 1 e. CC |
13 |
12
|
addid2i |
|- ( 0 + 1 ) = 1 |
14 |
|
ax-i2m1 |
|- ( ( _i x. _i ) + 1 ) = 0 |
15 |
13 14
|
breq12i |
|- ( ( 0 + 1 ) < ( ( _i x. _i ) + 1 ) <-> 1 < 0 ) |
16 |
11 15
|
bitri |
|- ( 0 < ( _i x. _i ) <-> 1 < 0 ) |
17 |
7 16
|
mtbir |
|- -. 0 < ( _i x. _i ) |
18 |
|
msqgt0 |
|- ( ( _i e. RR /\ _i =/= 0 ) -> 0 < ( _i x. _i ) ) |
19 |
18
|
ex |
|- ( _i e. RR -> ( _i =/= 0 -> 0 < ( _i x. _i ) ) ) |
20 |
19
|
necon1bd |
|- ( _i e. RR -> ( -. 0 < ( _i x. _i ) -> _i = 0 ) ) |
21 |
17 20
|
mpi |
|- ( _i e. RR -> _i = 0 ) |
22 |
2 21
|
mto |
|- -. _i e. RR |