Theorem fsummulc1

Description: A finite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005) (Revised by Mario Carneiro, 24-Apr-2014)

	Ref	Expression
Hypotheses	fsummulc2.1	$$⊢\phi \to A\in \mathrm{Fin}$$
	fsummulc2.2	$$⊢\phi \to C\in ℂ$$
	fsummulc2.3	$$⊢\left(\phi \wedge k\in A\right)\to B\in ℂ$$
Assertion	fsummulc1	$$⊢\phi \to \sum _{k\in A}BC=\sum _{k\in A}BC$$

Proof

Step	Hyp	Ref	Expression
1		fsummulc2.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fsummulc2.2	$$⊢\phi \to C\in ℂ$$
3		fsummulc2.3	$$⊢\left(\phi \wedge k\in A\right)\to B\in ℂ$$
4	1 2 3	fsummulc2	$$⊢\phi \to C\sum _{k\in A}B=\sum _{k\in A}CB$$
5	1 3	fsumcl	$$⊢\phi \to \sum _{k\in A}B\in ℂ$$
6	5 2	mulcomd	$$⊢\phi \to \sum _{k\in A}BC=C\sum _{k\in A}B$$
7	2	adantr	$$⊢\left(\phi \wedge k\in A\right)\to C\in ℂ$$
8	3 7	mulcomd	$$⊢\left(\phi \wedge k\in A\right)\to BC=CB$$
9	8	sumeq2dv	$$⊢\phi \to \sum _{k\in A}BC=\sum _{k\in A}CB$$
10	4 6 9	3eqtr4d	$$⊢\phi \to \sum _{k\in A}BC=\sum _{k\in A}BC$$