Theorem cjneg

Description: Complex conjugate of negative. (Contributed by NM, 27-Feb-2005) (Revised by Mario Carneiro, 14-Jul-2014)

	Ref	Expression
Assertion	cjneg	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ - 𝐴 ) = - ( ∗ ‘ 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	neg2subd	⊢ ( 𝐴 ∈ ℂ → ( - ( ℜ ‘ 𝐴 ) − - ( i · ( ℑ ‘ 𝐴 ) ) ) = ( ( i · ( ℑ ‘ 𝐴 ) ) − ( ℜ ‘ 𝐴 ) ) )
9		reneg	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ - 𝐴 ) = - ( ℜ ‘ 𝐴 ) )
10		imneg	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ - 𝐴 ) = - ( ℑ ‘ 𝐴 ) )
11	10	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( i · ( ℑ ‘ - 𝐴 ) ) = ( i · - ( ℑ ‘ 𝐴 ) ) )
12		mulneg2	⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · - ( ℑ ‘ 𝐴 ) ) = - ( i · ( ℑ ‘ 𝐴 ) ) )
13	3 5 12	sylancr	⊢ ( 𝐴 ∈ ℂ → ( i · - ( ℑ ‘ 𝐴 ) ) = - ( i · ( ℑ ‘ 𝐴 ) ) )
14	11 13	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( i · ( ℑ ‘ - 𝐴 ) ) = - ( i · ( ℑ ‘ 𝐴 ) ) )
15	9 14	oveq12d	⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ - 𝐴 ) − ( i · ( ℑ ‘ - 𝐴 ) ) ) = ( - ( ℜ ‘ 𝐴 ) − - ( i · ( ℑ ‘ 𝐴 ) ) ) )
16	2 7	negsubdi2d	⊢ ( 𝐴 ∈ ℂ → - ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) = ( ( i · ( ℑ ‘ 𝐴 ) ) − ( ℜ ‘ 𝐴 ) ) )
17	8 15 16	3eqtr4d	⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ - 𝐴 ) − ( i · ( ℑ ‘ - 𝐴 ) ) ) = - ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) )
18		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
19		remim	⊢ ( - 𝐴 ∈ ℂ → ( ∗ ‘ - 𝐴 ) = ( ( ℜ ‘ - 𝐴 ) − ( i · ( ℑ ‘ - 𝐴 ) ) ) )
20	18 19	syl	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ - 𝐴 ) = ( ( ℜ ‘ - 𝐴 ) − ( i · ( ℑ ‘ - 𝐴 ) ) ) )
21		remim	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) = ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) )
22	21	negeqd	⊢ ( 𝐴 ∈ ℂ → - ( ∗ ‘ 𝐴 ) = - ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) )
23	17 20 22	3eqtr4d	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ - 𝐴 ) = - ( ∗ ‘ 𝐴 ) )