Step |
Hyp |
Ref |
Expression |
1 |
|
rernegcl |
|- ( A e. RR -> ( 0 -R A ) e. RR ) |
2 |
|
rernegcl |
|- ( ( 0 -R A ) e. RR -> ( 0 -R ( 0 -R A ) ) e. RR ) |
3 |
1 2
|
syl |
|- ( A e. RR -> ( 0 -R ( 0 -R A ) ) e. RR ) |
4 |
|
id |
|- ( A e. RR -> A e. RR ) |
5 |
|
renegid |
|- ( A e. RR -> ( A + ( 0 -R A ) ) = 0 ) |
6 |
|
elre0re |
|- ( A e. RR -> 0 e. RR ) |
7 |
5 6
|
eqeltrd |
|- ( A e. RR -> ( A + ( 0 -R A ) ) e. RR ) |
8 |
|
readdid1 |
|- ( A e. RR -> ( A + 0 ) = A ) |
9 |
|
repncan3 |
|- ( ( ( 0 -R A ) e. RR /\ 0 e. RR ) -> ( ( 0 -R A ) + ( 0 -R ( 0 -R A ) ) ) = 0 ) |
10 |
1 6 9
|
syl2anc |
|- ( A e. RR -> ( ( 0 -R A ) + ( 0 -R ( 0 -R A ) ) ) = 0 ) |
11 |
10
|
oveq2d |
|- ( A e. RR -> ( A + ( ( 0 -R A ) + ( 0 -R ( 0 -R A ) ) ) ) = ( A + 0 ) ) |
12 |
|
readdid2 |
|- ( A e. RR -> ( 0 + A ) = A ) |
13 |
8 11 12
|
3eqtr4d |
|- ( A e. RR -> ( A + ( ( 0 -R A ) + ( 0 -R ( 0 -R A ) ) ) ) = ( 0 + A ) ) |
14 |
|
recn |
|- ( A e. RR -> A e. CC ) |
15 |
1
|
recnd |
|- ( A e. RR -> ( 0 -R A ) e. CC ) |
16 |
3
|
recnd |
|- ( A e. RR -> ( 0 -R ( 0 -R A ) ) e. CC ) |
17 |
14 15 16
|
addassd |
|- ( A e. RR -> ( ( A + ( 0 -R A ) ) + ( 0 -R ( 0 -R A ) ) ) = ( A + ( ( 0 -R A ) + ( 0 -R ( 0 -R A ) ) ) ) ) |
18 |
5
|
oveq1d |
|- ( A e. RR -> ( ( A + ( 0 -R A ) ) + A ) = ( 0 + A ) ) |
19 |
13 17 18
|
3eqtr4d |
|- ( A e. RR -> ( ( A + ( 0 -R A ) ) + ( 0 -R ( 0 -R A ) ) ) = ( ( A + ( 0 -R A ) ) + A ) ) |
20 |
|
readdcan |
|- ( ( ( 0 -R ( 0 -R A ) ) e. RR /\ A e. RR /\ ( A + ( 0 -R A ) ) e. RR ) -> ( ( ( A + ( 0 -R A ) ) + ( 0 -R ( 0 -R A ) ) ) = ( ( A + ( 0 -R A ) ) + A ) <-> ( 0 -R ( 0 -R A ) ) = A ) ) |
21 |
20
|
biimpa |
|- ( ( ( ( 0 -R ( 0 -R A ) ) e. RR /\ A e. RR /\ ( A + ( 0 -R A ) ) e. RR ) /\ ( ( A + ( 0 -R A ) ) + ( 0 -R ( 0 -R A ) ) ) = ( ( A + ( 0 -R A ) ) + A ) ) -> ( 0 -R ( 0 -R A ) ) = A ) |
22 |
3 4 7 19 21
|
syl31anc |
|- ( A e. RR -> ( 0 -R ( 0 -R A ) ) = A ) |