Description: A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | recld.1 | |- ( ph -> A e. CC ) |
|
| cjrebd.2 | |- ( ph -> ( * ` A ) = A ) |
||
| Assertion | cjrebd | |- ( ph -> A e. RR ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | recld.1 | |- ( ph -> A e. CC ) |
|
| 2 | cjrebd.2 | |- ( ph -> ( * ` A ) = A ) |
|
| 3 | cjreb | |- ( A e. CC -> ( A e. RR <-> ( * ` A ) = A ) ) |
|
| 4 | 1 3 | syl | |- ( ph -> ( A e. RR <-> ( * ` A ) = A ) ) |
| 5 | 2 4 | mpbird | |- ( ph -> A e. RR ) |