Theorem eluzsubi

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 ) )

Proof

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 ) )