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Metamath Proof Explorer
Description: The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999) (Revised by Mario Carneiro, 6-Nov-2013)
|
|
Ref |
Expression |
|
Assertion |
imval |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) = ( ℜ ‘ ( 𝐴 / i ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fvoveq1 |
⊢ ( 𝑥 = 𝐴 → ( ℜ ‘ ( 𝑥 / i ) ) = ( ℜ ‘ ( 𝐴 / i ) ) ) |
| 2 |
|
df-im |
⊢ ℑ = ( 𝑥 ∈ ℂ ↦ ( ℜ ‘ ( 𝑥 / i ) ) ) |
| 3 |
|
fvex |
⊢ ( ℜ ‘ ( 𝐴 / i ) ) ∈ V |
| 4 |
1 2 3
|
fvmpt |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) = ( ℜ ‘ ( 𝐴 / i ) ) ) |