Theorem cjmulvald

Description: A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	recld.1	|- ( ph -> A e. CC )
Assertion	cjmulvald	|- ( ph -> ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recld.1	|- ( ph -> A e. CC )
2		cjmulval	|- ( A e. CC -> ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )
3	1 2	syl	|- ( ph -> ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )