Step |
Hyp |
Ref |
Expression |
1 |
|
telfsumo.1 |
|- ( k = j -> A = B ) |
2 |
|
telfsumo.2 |
|- ( k = ( j + 1 ) -> A = C ) |
3 |
|
telfsumo.3 |
|- ( k = M -> A = D ) |
4 |
|
telfsumo.4 |
|- ( k = N -> A = E ) |
5 |
|
telfsumo.5 |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
6 |
|
telfsumo.6 |
|- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC ) |
7 |
1
|
negeqd |
|- ( k = j -> -u A = -u B ) |
8 |
2
|
negeqd |
|- ( k = ( j + 1 ) -> -u A = -u C ) |
9 |
3
|
negeqd |
|- ( k = M -> -u A = -u D ) |
10 |
4
|
negeqd |
|- ( k = N -> -u A = -u E ) |
11 |
6
|
negcld |
|- ( ( ph /\ k e. ( M ... N ) ) -> -u A e. CC ) |
12 |
7 8 9 10 5 11
|
telfsumo |
|- ( ph -> sum_ j e. ( M ..^ N ) ( -u B - -u C ) = ( -u D - -u E ) ) |
13 |
6
|
ralrimiva |
|- ( ph -> A. k e. ( M ... N ) A e. CC ) |
14 |
|
elfzofz |
|- ( j e. ( M ..^ N ) -> j e. ( M ... N ) ) |
15 |
1
|
eleq1d |
|- ( k = j -> ( A e. CC <-> B e. CC ) ) |
16 |
15
|
rspccva |
|- ( ( A. k e. ( M ... N ) A e. CC /\ j e. ( M ... N ) ) -> B e. CC ) |
17 |
13 14 16
|
syl2an |
|- ( ( ph /\ j e. ( M ..^ N ) ) -> B e. CC ) |
18 |
|
fzofzp1 |
|- ( j e. ( M ..^ N ) -> ( j + 1 ) e. ( M ... N ) ) |
19 |
2
|
eleq1d |
|- ( k = ( j + 1 ) -> ( A e. CC <-> C e. CC ) ) |
20 |
19
|
rspccva |
|- ( ( A. k e. ( M ... N ) A e. CC /\ ( j + 1 ) e. ( M ... N ) ) -> C e. CC ) |
21 |
13 18 20
|
syl2an |
|- ( ( ph /\ j e. ( M ..^ N ) ) -> C e. CC ) |
22 |
17 21
|
neg2subd |
|- ( ( ph /\ j e. ( M ..^ N ) ) -> ( -u B - -u C ) = ( C - B ) ) |
23 |
22
|
sumeq2dv |
|- ( ph -> sum_ j e. ( M ..^ N ) ( -u B - -u C ) = sum_ j e. ( M ..^ N ) ( C - B ) ) |
24 |
3
|
eleq1d |
|- ( k = M -> ( A e. CC <-> D e. CC ) ) |
25 |
|
eluzfz1 |
|- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) ) |
26 |
5 25
|
syl |
|- ( ph -> M e. ( M ... N ) ) |
27 |
24 13 26
|
rspcdva |
|- ( ph -> D e. CC ) |
28 |
4
|
eleq1d |
|- ( k = N -> ( A e. CC <-> E e. CC ) ) |
29 |
|
eluzfz2 |
|- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) ) |
30 |
5 29
|
syl |
|- ( ph -> N e. ( M ... N ) ) |
31 |
28 13 30
|
rspcdva |
|- ( ph -> E e. CC ) |
32 |
27 31
|
neg2subd |
|- ( ph -> ( -u D - -u E ) = ( E - D ) ) |
33 |
12 23 32
|
3eqtr3d |
|- ( ph -> sum_ j e. ( M ..^ N ) ( C - B ) = ( E - D ) ) |