Theorem angval

Description: Define the angle function, which takes two complex numbers, treated as vectors from the origin, and returns the angle between them, in the range ( -pi , pi ] . To convert from the geometry notation, m A B C , the measure of the angle with legs A B , C B where C is more counterclockwise for positive angles, is represented by ( ( C - B ) F ( A - B ) ) . (Contributed by Mario Carneiro, 23-Sep-2014)

	Ref	Expression
Hypothesis	ang.1	⊢ 𝐹 = ( 𝑥 ∈ ( ℂ ∖ { 0 } ) , 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( ℑ ‘ ( log ‘ ( 𝑦 / 𝑥 ) ) ) )
Assertion	angval	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 𝐹 𝐵 ) = ( ℑ ‘ ( log ‘ ( 𝐵 / 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ang.1	⊢ 𝐹 = ( 𝑥 ∈ ( ℂ ∖ { 0 } ) , 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( ℑ ‘ ( log ‘ ( 𝑦 / 𝑥 ) ) ) )
2		eldifsn	⊢ ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ↔ ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) )
3		eldifsn	⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 } ) ↔ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
4		oveq12	⊢ ( ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐴 ) → ( 𝑦 / 𝑥 ) = ( 𝐵 / 𝐴 ) )
5	4	ancoms	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑦 / 𝑥 ) = ( 𝐵 / 𝐴 ) )
6	5	fveq2d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( log ‘ ( 𝑦 / 𝑥 ) ) = ( log ‘ ( 𝐵 / 𝐴 ) ) )
7	6	fveq2d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ℑ ‘ ( log ‘ ( 𝑦 / 𝑥 ) ) ) = ( ℑ ‘ ( log ‘ ( 𝐵 / 𝐴 ) ) ) )
8		fvex	⊢ ( ℑ ‘ ( log ‘ ( 𝐵 / 𝐴 ) ) ) ∈ V
9	7 1 8	ovmpoa	⊢ ( ( 𝐴 ∈ ( ℂ ∖ { 0 } ) ∧ 𝐵 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐴 𝐹 𝐵 ) = ( ℑ ‘ ( log ‘ ( 𝐵 / 𝐴 ) ) ) )
10	2 3 9	syl2anbr	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 𝐹 𝐵 ) = ( ℑ ‘ ( log ‘ ( 𝐵 / 𝐴 ) ) ) )