Theorem imcj

Description: Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005) (Revised by Mario Carneiro, 14-Jul-2014)

	Ref	Expression
Assertion	imcj	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = - ( ℑ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		recl	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ )
2	1	recnd	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℂ )
3		ax-icn	⊢ i ∈ ℂ
4		imcl	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ )
5	4	recnd	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℂ )
6		mulcl	⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ )
7	3 5 6	sylancr	⊢ ( 𝐴 ∈ ℂ → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ )
8	2 7	negsubd	⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) + - ( i · ( ℑ ‘ 𝐴 ) ) ) = ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) )
9		mulneg2	⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · - ( ℑ ‘ 𝐴 ) ) = - ( i · ( ℑ ‘ 𝐴 ) ) )
10	3 5 9	sylancr	⊢ ( 𝐴 ∈ ℂ → ( i · - ( ℑ ‘ 𝐴 ) ) = - ( i · ( ℑ ‘ 𝐴 ) ) )
11	10	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) = ( ( ℜ ‘ 𝐴 ) + - ( i · ( ℑ ‘ 𝐴 ) ) ) )
12		remim	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) = ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) )
13	8 11 12	3eqtr4rd	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) = ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) )
14	13	fveq2d	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = ( ℑ ‘ ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) )
15	4	renegcld	⊢ ( 𝐴 ∈ ℂ → - ( ℑ ‘ 𝐴 ) ∈ ℝ )
16		crim	⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℝ ∧ - ( ℑ ‘ 𝐴 ) ∈ ℝ ) → ( ℑ ‘ ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) = - ( ℑ ‘ 𝐴 ) )
17	1 15 16	syl2anc	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) = - ( ℑ ‘ 𝐴 ) )
18	14 17	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = - ( ℑ ‘ 𝐴 ) )