Theorem climresd

Description: A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 23-Apr-2023)

	Ref	Expression
Hypotheses	climresd.1	|- ( ph -> M e. ZZ )
	climresd.2	|- ( ph -> F e. V )
Assertion	climresd	|- ( ph -> ( ( F |` ( ZZ>= ` M ) ) ~~> A <-> F ~~> A ) )

Proof

Step	Hyp	Ref	Expression
1		climresd.1	|- ( ph -> M e. ZZ )
2		climresd.2	|- ( ph -> F e. V )
3		climres	|- ( ( M e. ZZ /\ F e. V ) -> ( ( F |` ( ZZ>= ` M ) ) ~~> A <-> F ~~> A ) )
4	1 2 3	syl2anc	|- ( ph -> ( ( F |` ( ZZ>= ` M ) ) ~~> A <-> F ~~> A ) )