Metamath Proof Explorer

Theorem cjreb

Description: A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by NM, 2-Jul-2005) (Revised by Mario Carneiro, 14-Jul-2014)

Ref Expression
Assertion cjreb ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ ↔ ( ∗ ‘ 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 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 replim ⊢ ( 𝐴 ∈ ℂ → 𝐴 = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) )
15 13 14 eqeq12d ⊢ ( 𝐴 ∈ ℂ → ( ( ∗ ‘ 𝐴 ) = 𝐴 ↔ ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) )
16 5 negcld ⊢ ( 𝐴 ∈ ℂ → - ( ℑ ‘ 𝐴 ) ∈ ℂ )
17 mulcl ⊢ ( ( i ∈ ℂ ∧ - ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · - ( ℑ ‘ 𝐴 ) ) ∈ ℂ )
18 3 16 17 sylancr ⊢ ( 𝐴 ∈ ℂ → ( i · - ( ℑ ‘ 𝐴 ) ) ∈ ℂ )
19 2 18 7 addcand ⊢ ( 𝐴 ∈ ℂ → ( ( ( ℜ ‘ 𝐴 ) + ( i · - ( ℑ ‘ 𝐴 ) ) ) = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ↔ ( i · - ( ℑ ‘ 𝐴 ) ) = ( i · ( ℑ ‘ 𝐴 ) ) ) )
20 eqcom ⊢ ( - ( ℑ ‘ 𝐴 ) = ( ℑ ‘ 𝐴 ) ↔ ( ℑ ‘ 𝐴 ) = - ( ℑ ‘ 𝐴 ) )
21 5 eqnegd ⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ 𝐴 ) = - ( ℑ ‘ 𝐴 ) ↔ ( ℑ ‘ 𝐴 ) = 0 ) )
22 20 21 syl5bb ⊢ ( 𝐴 ∈ ℂ → ( - ( ℑ ‘ 𝐴 ) = ( ℑ ‘ 𝐴 ) ↔ ( ℑ ‘ 𝐴 ) = 0 ) )
23 ine0 ⊢ i ≠ 0
24 3 23 pm3.2i ⊢ ( i ∈ ℂ ∧ i ≠ 0 )
25 24 a1i ⊢ ( 𝐴 ∈ ℂ → ( i ∈ ℂ ∧ i ≠ 0 ) )
26 mulcan ⊢ ( ( - ( ℑ ‘ 𝐴 ) ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ∧ ( i ∈ ℂ ∧ i ≠ 0 ) ) → ( ( i · - ( ℑ ‘ 𝐴 ) ) = ( i · ( ℑ ‘ 𝐴 ) ) ↔ - ( ℑ ‘ 𝐴 ) = ( ℑ ‘ 𝐴 ) ) )
27 16 5 25 26 syl3anc ⊢ ( 𝐴 ∈ ℂ → ( ( i · - ( ℑ ‘ 𝐴 ) ) = ( i · ( ℑ ‘ 𝐴 ) ) ↔ - ( ℑ ‘ 𝐴 ) = ( ℑ ‘ 𝐴 ) ) )
28 reim0b ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ ↔ ( ℑ ‘ 𝐴 ) = 0 ) )
29 22 27 28 3bitr4d ⊢ ( 𝐴 ∈ ℂ → ( ( i · - ( ℑ ‘ 𝐴 ) ) = ( i · ( ℑ ‘ 𝐴 ) ) ↔ 𝐴 ∈ ℝ ) )
30 15 19 29 3bitrrd ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ ↔ ( ∗ ‘ 𝐴 ) = 𝐴 ) )