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	|- ( ph -> A e. Fin )
	fsummulc2.2	|- ( ph -> C e. CC )
	fsummulc2.3	|- ( ( ph /\ k e. A ) -> B e. CC )
Assertion	fsummulc1	|- ( ph -> ( sum_ k e. A B x. C ) = sum_ k e. A ( B x. C ) )

Proof

Step	Hyp	Ref	Expression
1		fsummulc2.1	|- ( ph -> A e. Fin )
2		fsummulc2.2	|- ( ph -> C e. CC )
3		fsummulc2.3	|- ( ( ph /\ k e. A ) -> B e. CC )
4	1 2 3	fsummulc2	|- ( ph -> ( C x. sum_ k e. A B ) = sum_ k e. A ( C x. B ) )
5	1 3	fsumcl	|- ( ph -> sum_ k e. A B e. CC )
6	5 2	mulcomd	|- ( ph -> ( sum_ k e. A B x. C ) = ( C x. sum_ k e. A B ) )
7	2	adantr	|- ( ( ph /\ k e. A ) -> C e. CC )
8	3 7	mulcomd	|- ( ( ph /\ k e. A ) -> ( B x. C ) = ( C x. B ) )
9	8	sumeq2dv	|- ( ph -> sum_ k e. A ( B x. C ) = sum_ k e. A ( C x. B ) )
10	4 6 9	3eqtr4d	|- ( ph -> ( sum_ k e. A B x. C ) = sum_ k e. A ( B x. C ) )