Metamath Proof Explorer
Description: Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
recld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
readdd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
Assertion |
immuld |
⊢ ( 𝜑 → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
recld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
readdd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
immul |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) ) ) |