Metamath Proof Explorer
Description: The imaginary part of a complex number representation. Definition
10-3.1 of Gleason p. 132. (Contributed by NM, 10-May-1999)
|
|
Ref |
Expression |
|
Hypotheses |
crre.1 |
⊢ 𝐴 ∈ ℝ |
|
|
crre.2 |
⊢ 𝐵 ∈ ℝ |
|
Assertion |
crimi |
⊢ ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐵 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
crre.1 |
⊢ 𝐴 ∈ ℝ |
2 |
|
crre.2 |
⊢ 𝐵 ∈ ℝ |
3 |
|
crim |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐵 ) |
4 |
1 2 3
|
mp2an |
⊢ ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐵 |