Description: If a complex number equals its square, it must be 0 or 1. (Contributed by NM, 6-Jun-2006)
Ref | Expression | ||
---|---|---|---|
Assertion | sq01 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne | |
|
2 | sqval | |
|
3 | mulid1 | |
|
4 | 3 | eqcomd | |
5 | 2 4 | eqeq12d | |
6 | 5 | adantr | |
7 | ax-1cn | |
|
8 | mulcan | |
|
9 | 7 8 | mp3an2 | |
10 | 9 | anabss5 | |
11 | 6 10 | bitrd | |
12 | 11 | biimpd | |
13 | 12 | impancom | |
14 | 1 13 | syl5bir | |
15 | 14 | orrd | |
16 | 15 | ex | |
17 | sq0 | |
|
18 | oveq1 | |
|
19 | id | |
|
20 | 17 18 19 | 3eqtr4a | |
21 | sq1 | |
|
22 | oveq1 | |
|
23 | id | |
|
24 | 21 22 23 | 3eqtr4a | |
25 | 20 24 | jaoi | |
26 | 16 25 | impbid1 | |