Metamath Proof Explorer


Theorem telfsum2

Description: Sum of a telescoping series. (Contributed by Mario Carneiro, 15-Jun-2014) (Revised by Mario Carneiro, 2-May-2016)

Ref Expression
Hypotheses telfsum.1
|- ( k = j -> A = B )
telfsum.2
|- ( k = ( j + 1 ) -> A = C )
telfsum.3
|- ( k = M -> A = D )
telfsum.4
|- ( k = ( N + 1 ) -> A = E )
telfsum.5
|- ( ph -> N e. ZZ )
telfsum.6
|- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
telfsum.7
|- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )
Assertion telfsum2
|- ( ph -> sum_ j e. ( M ... N ) ( C - B ) = ( E - D ) )

Proof

Step Hyp Ref Expression
1 telfsum.1
 |-  ( k = j -> A = B )
2 telfsum.2
 |-  ( k = ( j + 1 ) -> A = C )
3 telfsum.3
 |-  ( k = M -> A = D )
4 telfsum.4
 |-  ( k = ( N + 1 ) -> A = E )
5 telfsum.5
 |-  ( ph -> N e. ZZ )
6 telfsum.6
 |-  ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
7 telfsum.7
 |-  ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )
8 fzval3
 |-  ( N e. ZZ -> ( M ... N ) = ( M ..^ ( N + 1 ) ) )
9 5 8 syl
 |-  ( ph -> ( M ... N ) = ( M ..^ ( N + 1 ) ) )
10 9 sumeq1d
 |-  ( ph -> sum_ j e. ( M ... N ) ( C - B ) = sum_ j e. ( M ..^ ( N + 1 ) ) ( C - B ) )
11 1 2 3 4 6 7 telfsumo2
 |-  ( ph -> sum_ j e. ( M ..^ ( N + 1 ) ) ( C - B ) = ( E - D ) )
12 10 11 eqtrd
 |-  ( ph -> sum_ j e. ( M ... N ) ( C - B ) = ( E - D ) )