Theorem imre

Description: The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005) (Revised by Mario Carneiro, 6-Nov-2013)

	Ref	Expression
Assertion	imre	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) = ( ℜ ‘ ( - i · 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imval	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) = ( ℜ ‘ ( 𝐴 / i ) ) )
2		ax-icn	⊢ i ∈ ℂ
3		ine0	⊢ i ≠ 0
4		divrec2	⊢ ( ( 𝐴 ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0 ) → ( 𝐴 / i ) = ( ( 1 / i ) · 𝐴 ) )
5	2 3 4	mp3an23	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 / i ) = ( ( 1 / i ) · 𝐴 ) )
6		irec	⊢ ( 1 / i ) = - i
7	6	oveq1i	⊢ ( ( 1 / i ) · 𝐴 ) = ( - i · 𝐴 )
8	5 7	eqtrdi	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 / i ) = ( - i · 𝐴 ) )
9	8	fveq2d	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( 𝐴 / i ) ) = ( ℜ ‘ ( - i · 𝐴 ) ) )
10	1 9	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) = ( ℜ ‘ ( - i · 𝐴 ) ) )