Description: Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypothesis | abscld.1 | |- ( ph -> A e. CC ) |
|
Assertion | absvalsq2d | |- ( ph -> ( ( abs ` A ) ^ 2 ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 | |- ( ph -> A e. CC ) |
|
2 | absvalsq2 | |- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) |
|
3 | 1 2 | syl | |- ( ph -> ( ( abs ` A ) ^ 2 ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) |