Step |
Hyp |
Ref |
Expression |
1 |
|
reval |
|- ( A e. CC -> ( Re ` A ) = ( ( A + ( * ` A ) ) / 2 ) ) |
2 |
1
|
oveq2d |
|- ( A e. CC -> ( 2 x. ( Re ` A ) ) = ( 2 x. ( ( A + ( * ` A ) ) / 2 ) ) ) |
3 |
|
cjcl |
|- ( A e. CC -> ( * ` A ) e. CC ) |
4 |
|
addcl |
|- ( ( A e. CC /\ ( * ` A ) e. CC ) -> ( A + ( * ` A ) ) e. CC ) |
5 |
3 4
|
mpdan |
|- ( A e. CC -> ( A + ( * ` A ) ) e. CC ) |
6 |
|
2cn |
|- 2 e. CC |
7 |
|
2ne0 |
|- 2 =/= 0 |
8 |
|
divcan2 |
|- ( ( ( A + ( * ` A ) ) e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( 2 x. ( ( A + ( * ` A ) ) / 2 ) ) = ( A + ( * ` A ) ) ) |
9 |
6 7 8
|
mp3an23 |
|- ( ( A + ( * ` A ) ) e. CC -> ( 2 x. ( ( A + ( * ` A ) ) / 2 ) ) = ( A + ( * ` A ) ) ) |
10 |
5 9
|
syl |
|- ( A e. CC -> ( 2 x. ( ( A + ( * ` A ) ) / 2 ) ) = ( A + ( * ` A ) ) ) |
11 |
2 10
|
eqtr2d |
|- ( A e. CC -> ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) ) ) |