| Step |
Hyp |
Ref |
Expression |
| 1 |
|
reneg1lt0 |
|- ( 0 -R 1 ) < 0 |
| 2 |
|
1re |
|- 1 e. RR |
| 3 |
|
rernegcl |
|- ( 1 e. RR -> ( 0 -R 1 ) e. RR ) |
| 4 |
2 3
|
ax-mp |
|- ( 0 -R 1 ) e. RR |
| 5 |
|
0re |
|- 0 e. RR |
| 6 |
4 5
|
ltnsymi |
|- ( ( 0 -R 1 ) < 0 -> -. 0 < ( 0 -R 1 ) ) |
| 7 |
1 6
|
ax-mp |
|- -. 0 < ( 0 -R 1 ) |
| 8 |
|
reixi |
|- ( _i x. _i ) = ( 0 -R 1 ) |
| 9 |
8
|
breq2i |
|- ( 0 < ( _i x. _i ) <-> 0 < ( 0 -R 1 ) ) |
| 10 |
7 9
|
mtbir |
|- -. 0 < ( _i x. _i ) |
| 11 |
|
id |
|- ( _i e. RR -> _i e. RR ) |
| 12 |
|
0ne1 |
|- 0 =/= 1 |
| 13 |
12
|
a1i |
|- ( _i e. RR -> 0 =/= 1 ) |
| 14 |
|
id |
|- ( _i = 0 -> _i = 0 ) |
| 15 |
14 14
|
oveq12d |
|- ( _i = 0 -> ( _i x. _i ) = ( 0 x. 0 ) ) |
| 16 |
15
|
oveq1d |
|- ( _i = 0 -> ( ( _i x. _i ) + 1 ) = ( ( 0 x. 0 ) + 1 ) ) |
| 17 |
|
ax-i2m1 |
|- ( ( _i x. _i ) + 1 ) = 0 |
| 18 |
|
remul02 |
|- ( 0 e. RR -> ( 0 x. 0 ) = 0 ) |
| 19 |
5 18
|
ax-mp |
|- ( 0 x. 0 ) = 0 |
| 20 |
19
|
oveq1i |
|- ( ( 0 x. 0 ) + 1 ) = ( 0 + 1 ) |
| 21 |
|
readdlid |
|- ( 1 e. RR -> ( 0 + 1 ) = 1 ) |
| 22 |
2 21
|
ax-mp |
|- ( 0 + 1 ) = 1 |
| 23 |
20 22
|
eqtri |
|- ( ( 0 x. 0 ) + 1 ) = 1 |
| 24 |
16 17 23
|
3eqtr3g |
|- ( _i = 0 -> 0 = 1 ) |
| 25 |
24
|
adantl |
|- ( ( _i e. RR /\ _i = 0 ) -> 0 = 1 ) |
| 26 |
13 25
|
mteqand |
|- ( _i e. RR -> _i =/= 0 ) |
| 27 |
11 26
|
sn-msqgt0d |
|- ( _i e. RR -> 0 < ( _i x. _i ) ) |
| 28 |
10 27
|
mto |
|- -. _i e. RR |