Description: The integers are a closed set in the topology on CC . (Contributed by Mario Carneiro, 17-Feb-2015)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | recld2.1 | |- J = ( TopOpen ` CCfld ) |
|
| Assertion | zcld2 | |- ZZ e. ( Clsd ` J ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | recld2.1 | |- J = ( TopOpen ` CCfld ) |
|
| 2 | 1 | recld2 | |- RR e. ( Clsd ` J ) |
| 3 | 1 | tgioo2 | |- ( topGen ` ran (,) ) = ( J |`t RR ) |
| 4 | 3 | eqcomi | |- ( J |`t RR ) = ( topGen ` ran (,) ) |
| 5 | 4 | zcld | |- ZZ e. ( Clsd ` ( J |`t RR ) ) |
| 6 | restcldr | |- ( ( RR e. ( Clsd ` J ) /\ ZZ e. ( Clsd ` ( J |`t RR ) ) ) -> ZZ e. ( Clsd ` J ) ) |
|
| 7 | 2 5 6 | mp2an | |- ZZ e. ( Clsd ` J ) |