| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ax-1ne0 |
|- 1 =/= 0 |
| 2 |
|
1re |
|- 1 e. RR |
| 3 |
|
0re |
|- 0 e. RR |
| 4 |
2 3
|
lttri2i |
|- ( 1 =/= 0 <-> ( 1 < 0 \/ 0 < 1 ) ) |
| 5 |
1 4
|
mpbi |
|- ( 1 < 0 \/ 0 < 1 ) |
| 6 |
|
rernegcl |
|- ( 1 e. RR -> ( 0 -R 1 ) e. RR ) |
| 7 |
2 6
|
mp1i |
|- ( 1 < 0 -> ( 0 -R 1 ) e. RR ) |
| 8 |
|
relt0neg1 |
|- ( 1 e. RR -> ( 1 < 0 <-> 0 < ( 0 -R 1 ) ) ) |
| 9 |
2 8
|
ax-mp |
|- ( 1 < 0 <-> 0 < ( 0 -R 1 ) ) |
| 10 |
9
|
biimpi |
|- ( 1 < 0 -> 0 < ( 0 -R 1 ) ) |
| 11 |
7 7 10 10
|
mulgt0d |
|- ( 1 < 0 -> 0 < ( ( 0 -R 1 ) x. ( 0 -R 1 ) ) ) |
| 12 |
|
1red |
|- ( 1 e. RR -> 1 e. RR ) |
| 13 |
6 12
|
remulneg2d |
|- ( 1 e. RR -> ( ( 0 -R 1 ) x. ( 0 -R 1 ) ) = ( 0 -R ( ( 0 -R 1 ) x. 1 ) ) ) |
| 14 |
|
ax-1rid |
|- ( ( 0 -R 1 ) e. RR -> ( ( 0 -R 1 ) x. 1 ) = ( 0 -R 1 ) ) |
| 15 |
6 14
|
syl |
|- ( 1 e. RR -> ( ( 0 -R 1 ) x. 1 ) = ( 0 -R 1 ) ) |
| 16 |
15
|
oveq2d |
|- ( 1 e. RR -> ( 0 -R ( ( 0 -R 1 ) x. 1 ) ) = ( 0 -R ( 0 -R 1 ) ) ) |
| 17 |
|
renegneg |
|- ( 1 e. RR -> ( 0 -R ( 0 -R 1 ) ) = 1 ) |
| 18 |
13 16 17
|
3eqtrd |
|- ( 1 e. RR -> ( ( 0 -R 1 ) x. ( 0 -R 1 ) ) = 1 ) |
| 19 |
2 18
|
ax-mp |
|- ( ( 0 -R 1 ) x. ( 0 -R 1 ) ) = 1 |
| 20 |
11 19
|
breqtrdi |
|- ( 1 < 0 -> 0 < 1 ) |
| 21 |
|
id |
|- ( 0 < 1 -> 0 < 1 ) |
| 22 |
20 21
|
jaoi |
|- ( ( 1 < 0 \/ 0 < 1 ) -> 0 < 1 ) |
| 23 |
5 22
|
ax-mp |
|- 0 < 1 |