Description: If the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | liminfresuz2.1 | |- ( ph -> M e. ZZ ) |
|
| liminfresuz2.2 | |- Z = ( ZZ>= ` M ) |
||
| liminfresuz2.3 | |- ( ph -> F e. V ) |
||
| liminfresuz2.4 | |- ( ph -> dom F C_ ZZ ) |
||
| Assertion | liminfresuz2 | |- ( ph -> ( liminf ` ( F |` Z ) ) = ( liminf ` F ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | liminfresuz2.1 | |- ( ph -> M e. ZZ ) |
|
| 2 | liminfresuz2.2 | |- Z = ( ZZ>= ` M ) |
|
| 3 | liminfresuz2.3 | |- ( ph -> F e. V ) |
|
| 4 | liminfresuz2.4 | |- ( 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 | liminfresuz | |- ( ph -> ( liminf ` ( F |` Z ) ) = ( liminf ` F ) ) |