Metamath Proof Explorer


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 ) ) )