| Step |
Hyp |
Ref |
Expression |
| 1 |
|
renegid |
|- ( A e. RR -> ( A + ( 0 -R A ) ) = 0 ) |
| 2 |
1
|
oveq2d |
|- ( A e. RR -> ( ( 0 -R A ) + ( A + ( 0 -R A ) ) ) = ( ( 0 -R A ) + 0 ) ) |
| 3 |
|
rernegcl |
|- ( A e. RR -> ( 0 -R A ) e. RR ) |
| 4 |
|
readdrid |
|- ( ( 0 -R A ) e. RR -> ( ( 0 -R A ) + 0 ) = ( 0 -R A ) ) |
| 5 |
3 4
|
syl |
|- ( A e. RR -> ( ( 0 -R A ) + 0 ) = ( 0 -R A ) ) |
| 6 |
2 5
|
eqtrd |
|- ( A e. RR -> ( ( 0 -R A ) + ( A + ( 0 -R A ) ) ) = ( 0 -R A ) ) |
| 7 |
3
|
recnd |
|- ( A e. RR -> ( 0 -R A ) e. CC ) |
| 8 |
|
recn |
|- ( A e. RR -> A e. CC ) |
| 9 |
7 8 7
|
addassd |
|- ( A e. RR -> ( ( ( 0 -R A ) + A ) + ( 0 -R A ) ) = ( ( 0 -R A ) + ( A + ( 0 -R A ) ) ) ) |
| 10 |
|
readdlid |
|- ( ( 0 -R A ) e. RR -> ( 0 + ( 0 -R A ) ) = ( 0 -R A ) ) |
| 11 |
3 10
|
syl |
|- ( A e. RR -> ( 0 + ( 0 -R A ) ) = ( 0 -R A ) ) |
| 12 |
6 9 11
|
3eqtr4d |
|- ( A e. RR -> ( ( ( 0 -R A ) + A ) + ( 0 -R A ) ) = ( 0 + ( 0 -R A ) ) ) |
| 13 |
|
id |
|- ( A e. RR -> A e. RR ) |
| 14 |
3 13
|
readdcld |
|- ( A e. RR -> ( ( 0 -R A ) + A ) e. RR ) |
| 15 |
|
elre0re |
|- ( A e. RR -> 0 e. RR ) |
| 16 |
|
readdcan2 |
|- ( ( ( ( 0 -R A ) + A ) e. RR /\ 0 e. RR /\ ( 0 -R A ) e. RR ) -> ( ( ( ( 0 -R A ) + A ) + ( 0 -R A ) ) = ( 0 + ( 0 -R A ) ) <-> ( ( 0 -R A ) + A ) = 0 ) ) |
| 17 |
14 15 3 16
|
syl3anc |
|- ( A e. RR -> ( ( ( ( 0 -R A ) + A ) + ( 0 -R A ) ) = ( 0 + ( 0 -R A ) ) <-> ( ( 0 -R A ) + A ) = 0 ) ) |
| 18 |
12 17
|
mpbid |
|- ( A e. RR -> ( ( 0 -R A ) + A ) = 0 ) |