Theorem fsumzcl

Description: Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005) (Revised by Mario Carneiro, 22-Apr-2014)

	Ref	Expression
Hypotheses	fsumcl.1	|- ( ph -> A e. Fin )
	fsumzcl.2	|- ( ( ph /\ k e. A ) -> B e. ZZ )
Assertion	fsumzcl	|- ( ph -> sum_ k e. A B e. ZZ )

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	|- ( ph -> A e. Fin )
2		fsumzcl.2	|- ( ( ph /\ k e. A ) -> B e. ZZ )
3		zsscn	|- ZZ C_ CC
4	3	a1i	|- ( ph -> ZZ C_ CC )
5		zaddcl	|- ( ( x e. ZZ /\ y e. ZZ ) -> ( x + y ) e. ZZ )
6	5	adantl	|- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x + y ) e. ZZ )
7		0zd	|- ( ph -> 0 e. ZZ )
8	4 6 1 2 7	fsumcllem	|- ( ph -> sum_ k e. A B e. ZZ )