Theorem sigarim

Description: Signed area takes value in reals. (Contributed by Saveliy Skresanov, 19-Sep-2017)

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

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		simpl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐴 ∈ ℂ )
4	3	cjcld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ 𝐴 ) ∈ ℂ )
5		simpr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ )
6	4 5	mulcld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ 𝐴 ) · 𝐵 ) ∈ ℂ )
7	6	imcld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( ( ∗ ‘ 𝐴 ) · 𝐵 ) ) ∈ ℝ )
8	2 7	eqeltrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 𝐺 𝐵 ) ∈ ℝ )