Theorem fsumcl

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

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

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	|- ( ph -> A e. Fin )
2		fsumcl.2	|- ( ( ph /\ k e. A ) -> B e. CC )
3		ssidd	|- ( ph -> CC C_ CC )
4		addcl	|- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
5	4	adantl	|- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
6		0cnd	|- ( ph -> 0 e. CC )
7	3 5 1 2 6	fsumcllem	|- ( ph -> sum_ k e. A B e. CC )