Description: The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | abscld.1 | |- ( ph -> A e. CC ) |
|
| absne0d.2 | |- ( ph -> A =/= 0 ) |
||
| Assertion | absne0d | |- ( ph -> ( abs ` A ) =/= 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | abscld.1 | |- ( ph -> A e. CC ) |
|
| 2 | absne0d.2 | |- ( ph -> A =/= 0 ) |
|
| 3 | 1 | abs00ad | |- ( ph -> ( ( abs ` A ) = 0 <-> A = 0 ) ) |
| 4 | 3 | necon3bid | |- ( ph -> ( ( abs ` A ) =/= 0 <-> A =/= 0 ) ) |
| 5 | 2 4 | mpbird | |- ( ph -> ( abs ` A ) =/= 0 ) |