Step |
Hyp |
Ref |
Expression |
1 |
|
recl |
|- ( A e. CC -> ( Re ` A ) e. RR ) |
2 |
1
|
recnd |
|- ( A e. CC -> ( Re ` A ) e. CC ) |
3 |
|
abscl |
|- ( ( Re ` A ) e. CC -> ( abs ` ( Re ` A ) ) e. RR ) |
4 |
2 3
|
syl |
|- ( A e. CC -> ( abs ` ( Re ` A ) ) e. RR ) |
5 |
|
abscl |
|- ( A e. CC -> ( abs ` A ) e. RR ) |
6 |
|
leabs |
|- ( ( Re ` A ) e. RR -> ( Re ` A ) <_ ( abs ` ( Re ` A ) ) ) |
7 |
1 6
|
syl |
|- ( A e. CC -> ( Re ` A ) <_ ( abs ` ( Re ` A ) ) ) |
8 |
|
absrele |
|- ( A e. CC -> ( abs ` ( Re ` A ) ) <_ ( abs ` A ) ) |
9 |
1 4 5 7 8
|
letrd |
|- ( A e. CC -> ( Re ` A ) <_ ( abs ` A ) ) |