Theorem cjmulrcli

Description: A complex number times its conjugate is real. (Contributed by NM, 11-May-1999)

	Ref	Expression
Hypothesis	recl.1	⊢ 𝐴 ∈ ℂ
Assertion	cjmulrcli	⊢ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ

Proof

Step	Hyp	Ref	Expression
1		recl.1	⊢ 𝐴 ∈ ℂ
2		cjmulrcl	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ )
3	1 2	ax-mp	⊢ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ