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 ) )