Metamath Proof Explorer
Description: Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999)
|
|
Ref |
Expression |
|
Hypotheses |
recl.1 |
⊢ 𝐴 ∈ ℂ |
|
|
readdi.2 |
⊢ 𝐵 ∈ ℂ |
|
Assertion |
imaddi |
⊢ ( ℑ ‘ ( 𝐴 + 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
recl.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
readdi.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
imadd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 + 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ 𝐵 ) ) ) |
4 |
1 2 3
|
mp2an |
⊢ ( ℑ ‘ ( 𝐴 + 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ 𝐵 ) ) |