Theorem ser0

Description: The value of the partial sums in a zero-valued infinite series. (Contributed by Mario Carneiro, 31-Aug-2013) (Revised by Mario Carneiro, 15-Dec-2014)

	Ref	Expression
Hypothesis	ser0.1	|- Z = ( ZZ>= ` M )
Assertion	ser0	|- ( N e. Z -> ( seq M ( + , ( Z X. { 0 } ) ) ` N ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		ser0.1	|- Z = ( ZZ>= ` M )
2		00id	|- ( 0 + 0 ) = 0
3	2	a1i	|- ( N e. Z -> ( 0 + 0 ) = 0 )
4	1	eleq2i	|- ( N e. Z <-> N e. ( ZZ>= ` M ) )
5	4	biimpi	|- ( N e. Z -> N e. ( ZZ>= ` M ) )
6		0cn	|- 0 e. CC
7		elfzuz	|- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
8	7 1	eleqtrrdi	|- ( k e. ( M ... N ) -> k e. Z )
9	8	adantl	|- ( ( N e. Z /\ k e. ( M ... N ) ) -> k e. Z )
10		fvconst2g	|- ( ( 0 e. CC /\ k e. Z ) -> ( ( Z X. { 0 } ) ` k ) = 0 )
11	6 9 10	sylancr	|- ( ( N e. Z /\ k e. ( M ... N ) ) -> ( ( Z X. { 0 } ) ` k ) = 0 )
12	3 5 11	seqid3	|- ( N e. Z -> ( seq M ( + , ( Z X. { 0 } ) ) ` N ) = 0 )