Description: The reciprocal of a nonzero number is nonzero. (Contributed by NM, 9-Feb-2006) (Proof shortened by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | recne0 | |- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) =/= 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn | |- 1 e. CC |
|
2 | ax-1ne0 | |- 1 =/= 0 |
|
3 | divne0 | |- ( ( ( 1 e. CC /\ 1 =/= 0 ) /\ ( A e. CC /\ A =/= 0 ) ) -> ( 1 / A ) =/= 0 ) |
|
4 | 1 2 3 | mpanl12 | |- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) =/= 0 ) |