Theorem cjmulge0d

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

	Ref	Expression
Hypothesis	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	cjmulge0d	⊢ ( 𝜑 → 0 ≤ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cjmulge0	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )
3	1 2	syl	⊢ ( 𝜑 → 0 ≤ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )