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Metamath Proof Explorer

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 )