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 ) ) ) |
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 ) ) ) |