Theorem fsumrecl

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

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

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	|- ( ph -> A e. Fin )
2		fsumrecl.2	|- ( ( ph /\ k e. A ) -> B e. RR )
3		ax-resscn	|- RR C_ CC
4	3	a1i	|- ( ph -> RR C_ CC )
5		readdcl	|- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR )
6	5	adantl	|- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
7		0red	|- ( ph -> 0 e. RR )
8	4 6 1 2 7	fsumcllem	|- ( ph -> sum_ k e. A B e. RR )