Theorem cjmulrcld

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

	Ref	Expression
Hypothesis	recld.1	|- ( ph -> A e. CC )
Assertion	cjmulrcld	|- ( ph -> ( A x. ( * ` A ) ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		recld.1	|- ( ph -> A e. CC )
2		cjmulrcl	|- ( A e. CC -> ( A x. ( * ` A ) ) e. RR )
3	1 2	syl	|- ( ph -> ( A x. ( * ` A ) ) e. RR )