Metamath Proof Explorer


Theorem absvalsq2d

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 ) ) )

Proof

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 ) ) )