Theorem isummulc1

Description: An infinite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005) (Revised by Mario Carneiro, 23-Apr-2014)

	Ref	Expression
Hypotheses	isumcl.1	$$⊢Z={\mathbb{Z}}_{\ge M}$$
	isumcl.2	$$⊢\phi \to M\in \mathbb{Z}$$
	isumcl.3	$$⊢\left(\phi \wedge k\in Z\right)\to F\left(k\right)=A$$
	isumcl.4	$$⊢\left(\phi \wedge k\in Z\right)\to A\in ℂ$$
	isumcl.5	$$⊢\phi \to seqM\left(+,F\right)\in \mathrm{dom}⇝$$
	summulc.6	$$⊢\phi \to B\in ℂ$$
Assertion	isummulc1	$$⊢\phi \to \sum _{k\in Z}AB=\sum _{k\in Z}AB$$

Proof

Step	Hyp	Ref	Expression
1		isumcl.1	$$⊢Z={\mathbb{Z}}_{\ge M}$$
2		isumcl.2	$$⊢\phi \to M\in \mathbb{Z}$$
3		isumcl.3	$$⊢\left(\phi \wedge k\in Z\right)\to F\left(k\right)=A$$
4		isumcl.4	$$⊢\left(\phi \wedge k\in Z\right)\to A\in ℂ$$
5		isumcl.5	$$⊢\phi \to seqM\left(+,F\right)\in \mathrm{dom}⇝$$
6		summulc.6	$$⊢\phi \to B\in ℂ$$
7	1 2 3 4 5 6	isummulc2	$$⊢\phi \to B\sum _{k\in Z}A=\sum _{k\in Z}BA$$
8	1 2 3 4 5	isumcl	$$⊢\phi \to \sum _{k\in Z}A\in ℂ$$
9	8 6	mulcomd	$$⊢\phi \to \sum _{k\in Z}AB=B\sum _{k\in Z}A$$
10	6	adantr	$$⊢\left(\phi \wedge k\in Z\right)\to B\in ℂ$$
11	4 10	mulcomd	$$⊢\left(\phi \wedge k\in Z\right)\to AB=BA$$
12	11	sumeq2dv	$$⊢\phi \to \sum _{k\in Z}AB=\sum _{k\in Z}BA$$
13	7 9 12	3eqtr4d	$$⊢\phi \to \sum _{k\in Z}AB=\sum _{k\in Z}AB$$