Description: Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007) (Proof shortened by SN, 7-Feb-2025)
Ref | Expression | ||
---|---|---|---|
Hypotheses | eluzsubi.1 | |- M e. ZZ |
|
eluzsubi.2 | |- K e. ZZ |
||
Assertion | eluzsubi | |- ( N e. ( ZZ>= ` ( M + K ) ) -> ( N - K ) e. ( ZZ>= ` M ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzsubi.1 | |- M e. ZZ |
|
2 | eluzsubi.2 | |- K e. ZZ |
|
3 | eluzsub | |- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( N - K ) e. ( ZZ>= ` M ) ) |
|
4 | 1 2 3 | mp3an12 | |- ( N e. ( ZZ>= ` ( M + K ) ) -> ( N - K ) e. ( ZZ>= ` M ) ) |