Theorem sigarac

Description: Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017)

	Ref	Expression
Hypothesis	sigar	⊢ 𝐺 = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( ℑ ‘ ( ( ∗ ‘ 𝑥 ) · 𝑦 ) ) )
Assertion	sigarac	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 𝐺 𝐵 ) = - ( 𝐵 𝐺 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		sigar	⊢ 𝐺 = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( ℑ ‘ ( ( ∗ ‘ 𝑥 ) · 𝑦 ) ) )
2	1	sigarval	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 𝐺 𝐵 ) = ( ℑ ‘ ( ( ∗ ‘ 𝐴 ) · 𝐵 ) ) )
3		cjcl	⊢ ( 𝐵 ∈ ℂ → ( ∗ ‘ 𝐵 ) ∈ ℂ )
4	3	adantl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ 𝐵 ) ∈ ℂ )
5		simpl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐴 ∈ ℂ )
6	4 5	cjmuld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( ( ∗ ‘ 𝐵 ) · 𝐴 ) ) = ( ( ∗ ‘ ( ∗ ‘ 𝐵 ) ) · ( ∗ ‘ 𝐴 ) ) )
7		simpr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ )
8	7	cjcjd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( ∗ ‘ 𝐵 ) ) = 𝐵 )
9	8	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ ( ∗ ‘ 𝐵 ) ) · ( ∗ ‘ 𝐴 ) ) = ( 𝐵 · ( ∗ ‘ 𝐴 ) ) )
10	5	cjcld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ 𝐴 ) ∈ ℂ )
11	7 10	mulcomd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 · ( ∗ ‘ 𝐴 ) ) = ( ( ∗ ‘ 𝐴 ) · 𝐵 ) )
12	6 9 11	3eqtrrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ 𝐴 ) · 𝐵 ) = ( ∗ ‘ ( ( ∗ ‘ 𝐵 ) · 𝐴 ) ) )
13	12	fveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( ( ∗ ‘ 𝐴 ) · 𝐵 ) ) = ( ℑ ‘ ( ∗ ‘ ( ( ∗ ‘ 𝐵 ) · 𝐴 ) ) ) )
14	4 5	mulcld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ 𝐵 ) · 𝐴 ) ∈ ℂ )
15	14	imcjd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( ∗ ‘ ( ( ∗ ‘ 𝐵 ) · 𝐴 ) ) ) = - ( ℑ ‘ ( ( ∗ ‘ 𝐵 ) · 𝐴 ) ) )
16	2 13 15	3eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 𝐺 𝐵 ) = - ( ℑ ‘ ( ( ∗ ‘ 𝐵 ) · 𝐴 ) ) )
17	1	sigarval	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 𝐺 𝐴 ) = ( ℑ ‘ ( ( ∗ ‘ 𝐵 ) · 𝐴 ) ) )
18	17	ancoms	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 𝐺 𝐴 ) = ( ℑ ‘ ( ( ∗ ‘ 𝐵 ) · 𝐴 ) ) )
19	18	negeqd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐵 𝐺 𝐴 ) = - ( ℑ ‘ ( ( ∗ ‘ 𝐵 ) · 𝐴 ) ) )
20	16 19	eqtr4d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 𝐺 𝐵 ) = - ( 𝐵 𝐺 𝐴 ) )