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Metamath Proof Explorer
Description: A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypothesis |
recld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
Assertion |
cjmulvald |
⊢ ( 𝜑 → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
recld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
| 2 |
|
cjmulval |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 3 |
1 2
|
syl |
⊢ ( 𝜑 → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |