Step |
Hyp |
Ref |
Expression |
1 |
|
2times |
|- ( A e. CC -> ( 2 x. A ) = ( A + A ) ) |
2 |
1
|
oveq2d |
|- ( A e. CC -> ( A - ( 2 x. A ) ) = ( A - ( A + A ) ) ) |
3 |
|
id |
|- ( A e. CC -> A e. CC ) |
4 |
3 3
|
addcld |
|- ( A e. CC -> ( A + A ) e. CC ) |
5 |
3 4
|
negsubd |
|- ( A e. CC -> ( A + -u ( A + A ) ) = ( A - ( A + A ) ) ) |
6 |
3 3
|
negdid |
|- ( A e. CC -> -u ( A + A ) = ( -u A + -u A ) ) |
7 |
6
|
oveq2d |
|- ( A e. CC -> ( A + -u ( A + A ) ) = ( A + ( -u A + -u A ) ) ) |
8 |
|
negcl |
|- ( A e. CC -> -u A e. CC ) |
9 |
3 8 8
|
addassd |
|- ( A e. CC -> ( ( A + -u A ) + -u A ) = ( A + ( -u A + -u A ) ) ) |
10 |
|
negid |
|- ( A e. CC -> ( A + -u A ) = 0 ) |
11 |
10
|
oveq1d |
|- ( A e. CC -> ( ( A + -u A ) + -u A ) = ( 0 + -u A ) ) |
12 |
8
|
addid2d |
|- ( A e. CC -> ( 0 + -u A ) = -u A ) |
13 |
11 12
|
eqtrd |
|- ( A e. CC -> ( ( A + -u A ) + -u A ) = -u A ) |
14 |
7 9 13
|
3eqtr2d |
|- ( A e. CC -> ( A + -u ( A + A ) ) = -u A ) |
15 |
2 5 14
|
3eqtr2d |
|- ( A e. CC -> ( A - ( 2 x. A ) ) = -u A ) |