Theorem cjmulge0

Description: A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005) (Revised by Mario Carneiro, 14-Jul-2014)

	Ref	Expression
Assertion	cjmulge0	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recl	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ )
2	1	resqcld	⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ∈ ℝ )
3		imcl	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ )
4	3	resqcld	⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ∈ ℝ )
5	1	sqge0d	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( ℜ ‘ 𝐴 ) ↑ 2 ) )
6	3	sqge0d	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( ℑ ‘ 𝐴 ) ↑ 2 ) )
7	2 4 5 6	addge0d	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) )
8		cjmulval	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) )
9	7 8	breqtrrd	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )