Step |
Hyp |
Ref |
Expression |
1 |
|
sigar |
⊢ 𝐺 = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( ℑ ‘ ( ( ∗ ‘ 𝑥 ) · 𝑦 ) ) ) |
2 |
1
|
sigarval |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐴 𝐺 𝐴 ) = ( ℑ ‘ ( ( ∗ ‘ 𝐴 ) · 𝐴 ) ) ) |
3 |
2
|
anidms |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 𝐺 𝐴 ) = ( ℑ ‘ ( ( ∗ ‘ 𝐴 ) · 𝐴 ) ) ) |
4 |
|
cjcl |
⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) ∈ ℂ ) |
5 |
|
id |
⊢ ( 𝐴 ∈ ℂ → 𝐴 ∈ ℂ ) |
6 |
4 5
|
mulcomd |
⊢ ( 𝐴 ∈ ℂ → ( ( ∗ ‘ 𝐴 ) · 𝐴 ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
7 |
|
cjmulrcl |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ ) |
8 |
6 7
|
eqeltrd |
⊢ ( 𝐴 ∈ ℂ → ( ( ∗ ‘ 𝐴 ) · 𝐴 ) ∈ ℝ ) |
9 |
8
|
reim0d |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ( ∗ ‘ 𝐴 ) · 𝐴 ) ) = 0 ) |
10 |
3 9
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 𝐺 𝐴 ) = 0 ) |