Metamath Proof Explorer


Theorem eluzaddi

Description: Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007) Shorten and remove M e. ZZ hypothesis. (Revised by SN, 7-Feb-2025)

Ref Expression
Hypothesis eluzaddi.1
|- K e. ZZ
Assertion eluzaddi
|- ( N e. ( ZZ>= ` M ) -> ( N + K ) e. ( ZZ>= ` ( M + K ) ) )

Proof

Step Hyp Ref Expression
1 eluzaddi.1
 |-  K e. ZZ
2 eluzadd
 |-  ( ( N e. ( ZZ>= ` M ) /\ K e. ZZ ) -> ( N + K ) e. ( ZZ>= ` ( M + K ) ) )
3 1 2 mpan2
 |-  ( N e. ( ZZ>= ` M ) -> ( N + K ) e. ( ZZ>= ` ( M + K ) ) )