Theorem sigarval

Description: Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017)

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

Proof

Step	Hyp	Ref	Expression
1		sigar	⊢ 𝐺 = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( ℑ ‘ ( ( ∗ ‘ 𝑥 ) · 𝑦 ) ) )
2		simpl	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑥 = 𝐴 )
3	2	fveq2d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∗ ‘ 𝑥 ) = ( ∗ ‘ 𝐴 ) )
4		simpr	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 )
5	3 4	oveq12d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ( ∗ ‘ 𝑥 ) · 𝑦 ) = ( ( ∗ ‘ 𝐴 ) · 𝐵 ) )
6	5	fveq2d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ℑ ‘ ( ( ∗ ‘ 𝑥 ) · 𝑦 ) ) = ( ℑ ‘ ( ( ∗ ‘ 𝐴 ) · 𝐵 ) ) )
7		fvex	⊢ ( ℑ ‘ ( ( ∗ ‘ 𝐴 ) · 𝐵 ) ) ∈ V
8	6 1 7	ovmpoa	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 𝐺 𝐵 ) = ( ℑ ‘ ( ( ∗ ‘ 𝐴 ) · 𝐵 ) ) )