Theorem fsumnn0cl

Description: Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015)

	Ref	Expression
Hypotheses	fsumcl.1	|- ( ph -> A e. Fin )
	fsumnn0cl.2	|- ( ( ph /\ k e. A ) -> B e. NN0 )
Assertion	fsumnn0cl	|- ( ph -> sum_ k e. A B e. NN0 )

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	|- ( ph -> A e. Fin )
2		fsumnn0cl.2	|- ( ( ph /\ k e. A ) -> B e. NN0 )
3		nn0sscn	|- NN0 C_ CC
4	3	a1i	|- ( ph -> NN0 C_ CC )
5		nn0addcl	|- ( ( x e. NN0 /\ y e. NN0 ) -> ( x + y ) e. NN0 )
6	5	adantl	|- ( ( ph /\ ( x e. NN0 /\ y e. NN0 ) ) -> ( x + y ) e. NN0 )
7		0nn0	|- 0 e. NN0
8	7	a1i	|- ( ph -> 0 e. NN0 )
9	4 6 1 2 8	fsumcllem	|- ( ph -> sum_ k e. A B e. NN0 )