Metamath Proof Explorer


Theorem recj

Description: Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014)

Ref Expression
Assertion recj ( 𝐴 ∈ ℂ → ( ℜ ‘ ( ∗ ‘ 𝐴 ) ) = ( ℜ ‘ 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 → ( ℜ ‘ ( ∗ ‘ 𝐴 ) ) = ( ℜ ‘ ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) )
15 4 renegcld ( 𝐴 ∈ ℂ → - ( ℑ ‘ 𝐴 ) ∈ ℝ )
16 crre ( ( ( ℜ ‘ 𝐴 ) ∈ ℝ ∧ - ( ℑ ‘ 𝐴 ) ∈ ℝ ) → ( ℜ ‘ ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) = ( ℜ ‘ 𝐴 ) )
17 1 15 16 syl2anc ( 𝐴 ∈ ℂ → ( ℜ ‘ ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) ) = ( ℜ ‘ 𝐴 ) )
18 14 17 eqtrd ( 𝐴 ∈ ℂ → ( ℜ ‘ ( ∗ ‘ 𝐴 ) ) = ( ℜ ‘ 𝐴 ) )