Theorem fsumneg

Description: Negation of a finite sum. (Contributed by Scott Fenton, 12-Jun-2013) (Revised by Mario Carneiro, 24-Apr-2014)

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

Proof

Step	Hyp	Ref	Expression
1		fsumneg.1	|- ( ph -> A e. Fin )
2		fsumneg.2	|- ( ( ph /\ k e. A ) -> B e. CC )
3		neg1cn	|- -u 1 e. CC
4	3	a1i	|- ( ph -> -u 1 e. CC )
5	1 4 2	fsummulc2	|- ( ph -> ( -u 1 x. sum_ k e. A B ) = sum_ k e. A ( -u 1 x. B ) )
6	1 2	fsumcl	|- ( ph -> sum_ k e. A B e. CC )
7	6	mulm1d	|- ( ph -> ( -u 1 x. sum_ k e. A B ) = -u sum_ k e. A B )
8	2	mulm1d	|- ( ( ph /\ k e. A ) -> ( -u 1 x. B ) = -u B )
9	8	sumeq2dv	|- ( ph -> sum_ k e. A ( -u 1 x. B ) = sum_ k e. A -u B )
10	5 7 9	3eqtr3rd	|- ( ph -> sum_ k e. A -u B = -u sum_ k e. A B )