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Metamath Proof Explorer

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 ) )