Description: If the domain of a function is a subset of the integers, the superior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | limsupresuz2.1 | |- ( ph -> M e. ZZ ) |
|
| limsupresuz2.2 | |- Z = ( ZZ>= ` M ) |
||
| limsupresuz2.3 | |- ( ph -> F e. V ) |
||
| limsupresuz2.4 | |- ( ph -> dom F C_ ZZ ) |
||
| Assertion | limsupresuz2 | |- ( ph -> ( limsup ` ( F |` Z ) ) = ( limsup ` F ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | limsupresuz2.1 | |- ( ph -> M e. ZZ ) |
|
| 2 | limsupresuz2.2 | |- Z = ( ZZ>= ` M ) |
|
| 3 | limsupresuz2.3 | |- ( ph -> F e. V ) |
|
| 4 | limsupresuz2.4 | |- ( ph -> dom F C_ ZZ ) |
|
| 5 | dmresss | |- dom ( F |` RR ) C_ dom F |
|
| 6 | 5 | a1i | |- ( ph -> dom ( F |` RR ) C_ dom F ) |
| 7 | 6 4 | sstrd | |- ( ph -> dom ( F |` RR ) C_ ZZ ) |
| 8 | 1 2 3 7 | limsupresuz | |- ( ph -> ( limsup ` ( F |` Z ) ) = ( limsup ` F ) ) |