Theorem climres

Description: A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013) (Revised by Mario Carneiro, 31-Jan-2014)

	Ref	Expression
Assertion	climres	|- ( ( M e. ZZ /\ F e. V ) -> ( ( F |` ( ZZ>= ` M ) ) ~~> A <-> F ~~> A ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		resexg	|- ( F e. V -> ( F |` ( ZZ>= ` M ) ) e. _V )
3	2	adantl	|- ( ( M e. ZZ /\ F e. V ) -> ( F |` ( ZZ>= ` M ) ) e. _V )
4		simpr	|- ( ( M e. ZZ /\ F e. V ) -> F e. V )
5		simpl	|- ( ( M e. ZZ /\ F e. V ) -> M e. ZZ )
6		fvres	|- ( k e. ( ZZ>= ` M ) -> ( ( F |` ( ZZ>= ` M ) ) ` k ) = ( F ` k ) )
7	6	adantl	|- ( ( ( M e. ZZ /\ F e. V ) /\ k e. ( ZZ>= ` M ) ) -> ( ( F |` ( ZZ>= ` M ) ) ` k ) = ( F ` k ) )
8	1 3 4 5 7	climeq	|- ( ( M e. ZZ /\ F e. V ) -> ( ( F |` ( ZZ>= ` M ) ) ~~> A <-> F ~~> A ) )