Step |
Hyp |
Ref |
Expression |
1 |
|
sigarimcd.sigar |
⊢ 𝐺 = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( ℑ ‘ ( ( ∗ ‘ 𝑥 ) · 𝑦 ) ) ) |
2 |
|
sigarimcd.a |
⊢ ( 𝜑 → ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ) |
3 |
|
sigariz.a |
⊢ ( 𝜑 → ( 𝐴 𝐺 𝐵 ) = 0 ) |
4 |
1
|
sigarac |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 𝐺 𝐵 ) = - ( 𝐵 𝐺 𝐴 ) ) |
5 |
2 4
|
syl |
⊢ ( 𝜑 → ( 𝐴 𝐺 𝐵 ) = - ( 𝐵 𝐺 𝐴 ) ) |
6 |
3 5
|
eqtr3d |
⊢ ( 𝜑 → 0 = - ( 𝐵 𝐺 𝐴 ) ) |
7 |
6
|
negeqd |
⊢ ( 𝜑 → - 0 = - - ( 𝐵 𝐺 𝐴 ) ) |
8 |
|
neg0 |
⊢ - 0 = 0 |
9 |
8
|
a1i |
⊢ ( 𝜑 → - 0 = 0 ) |
10 |
2
|
ancomd |
⊢ ( 𝜑 → ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) ) |
11 |
1 10
|
sigarimcd |
⊢ ( 𝜑 → ( 𝐵 𝐺 𝐴 ) ∈ ℂ ) |
12 |
11
|
negnegd |
⊢ ( 𝜑 → - - ( 𝐵 𝐺 𝐴 ) = ( 𝐵 𝐺 𝐴 ) ) |
13 |
7 9 12
|
3eqtr3rd |
⊢ ( 𝜑 → ( 𝐵 𝐺 𝐴 ) = 0 ) |