Description: Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007)
Ref | Expression | ||
---|---|---|---|
Hypotheses | absvalsqi.1 | |- A e. CC |
|
abssub.2 | |- B e. CC |
||
Assertion | sqabssubi | |- ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absvalsqi.1 | |- A e. CC |
|
2 | abssub.2 | |- B e. CC |
|
3 | sqabssub | |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) ) |
|
4 | 1 2 3 | mp2an | |- ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) |