Theorem imneg

Description: The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005) (Revised by Mario Carneiro, 14-Jul-2014)

	Ref	Expression
Assertion	imneg	⊢ ( 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	negdid	⊢ ( 𝐴 ∈ ℂ → - ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) = ( - ( ℜ ‘ 𝐴 ) + - ( i · ( ℑ ‘ 𝐴 ) ) ) )
9		replim	⊢ ( 𝐴 ∈ ℂ → 𝐴 = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) )
10	9	negeqd	⊢ ( 𝐴 ∈ ℂ → - 𝐴 = - ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) )
11		mulneg2	⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · - ( ℑ ‘ 𝐴 ) ) = - ( i · ( ℑ ‘ 𝐴 ) ) )
12	3 5 11	sylancr	⊢ ( 𝐴 ∈ ℂ → ( i · - ( ℑ ‘ 𝐴 ) ) = - ( i · ( ℑ ‘ 𝐴 ) ) )
13	12	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( - ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) = ( - ( ℜ ‘ 𝐴 ) + - ( i · ( ℑ ‘ 𝐴 ) ) ) )
14	8 10 13	3eqtr4d	⊢ ( 𝐴 ∈ ℂ → - 𝐴 = ( - ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) )
15	14	fveq2d	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ - 𝐴 ) = ( ℑ ‘ ( - ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) )
16	1	renegcld	⊢ ( 𝐴 ∈ ℂ → - ( ℜ ‘ 𝐴 ) ∈ ℝ )
17	4	renegcld	⊢ ( 𝐴 ∈ ℂ → - ( ℑ ‘ 𝐴 ) ∈ ℝ )
18		crim	⊢ ( ( - ( ℜ ‘ 𝐴 ) ∈ ℝ ∧ - ( ℑ ‘ 𝐴 ) ∈ ℝ ) → ( ℑ ‘ ( - ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) = - ( ℑ ‘ 𝐴 ) )
19	16 17 18	syl2anc	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( - ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) = - ( ℑ ‘ 𝐴 ) )
20	15 19	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ - 𝐴 ) = - ( ℑ ‘ 𝐴 ) )