Step |
Hyp |
Ref |
Expression |
1 |
|
0re |
|- 0 e. RR |
2 |
|
rennncan2 |
|- ( ( 0 e. RR /\ 0 e. RR /\ 0 e. RR ) -> ( ( 0 -R 0 ) -R ( 0 -R 0 ) ) = ( 0 -R 0 ) ) |
3 |
1 1 1 2
|
mp3an |
|- ( ( 0 -R 0 ) -R ( 0 -R 0 ) ) = ( 0 -R 0 ) |
4 |
|
re1m1e0m0 |
|- ( 1 -R 1 ) = ( 0 -R 0 ) |
5 |
3 4
|
eqtr4i |
|- ( ( 0 -R 0 ) -R ( 0 -R 0 ) ) = ( 1 -R 1 ) |
6 |
|
rernegcl |
|- ( 0 e. RR -> ( 0 -R 0 ) e. RR ) |
7 |
1 6
|
ax-mp |
|- ( 0 -R 0 ) e. RR |
8 |
|
sn-00idlem1 |
|- ( ( 0 -R 0 ) e. RR -> ( ( 0 -R 0 ) x. ( 0 -R 0 ) ) = ( ( 0 -R 0 ) -R ( 0 -R 0 ) ) ) |
9 |
7 8
|
ax-mp |
|- ( ( 0 -R 0 ) x. ( 0 -R 0 ) ) = ( ( 0 -R 0 ) -R ( 0 -R 0 ) ) |
10 |
|
1re |
|- 1 e. RR |
11 |
|
sn-00idlem1 |
|- ( 1 e. RR -> ( 1 x. ( 0 -R 0 ) ) = ( 1 -R 1 ) ) |
12 |
10 11
|
ax-mp |
|- ( 1 x. ( 0 -R 0 ) ) = ( 1 -R 1 ) |
13 |
5 9 12
|
3eqtr4i |
|- ( ( 0 -R 0 ) x. ( 0 -R 0 ) ) = ( 1 x. ( 0 -R 0 ) ) |
14 |
7
|
a1i |
|- ( ( 0 -R 0 ) =/= 0 -> ( 0 -R 0 ) e. RR ) |
15 |
|
1red |
|- ( ( 0 -R 0 ) =/= 0 -> 1 e. RR ) |
16 |
|
id |
|- ( ( 0 -R 0 ) =/= 0 -> ( 0 -R 0 ) =/= 0 ) |
17 |
14 15 14 16
|
remulcan2d |
|- ( ( 0 -R 0 ) =/= 0 -> ( ( ( 0 -R 0 ) x. ( 0 -R 0 ) ) = ( 1 x. ( 0 -R 0 ) ) <-> ( 0 -R 0 ) = 1 ) ) |
18 |
13 17
|
mpbii |
|- ( ( 0 -R 0 ) =/= 0 -> ( 0 -R 0 ) = 1 ) |