Step |
Hyp |
Ref |
Expression |
1 |
|
fsumrev.1 |
|- ( ph -> K e. ZZ ) |
2 |
|
fsumrev.2 |
|- ( ph -> M e. ZZ ) |
3 |
|
fsumrev.3 |
|- ( ph -> N e. ZZ ) |
4 |
|
fsumrev.4 |
|- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC ) |
5 |
|
fsumshftm.5 |
|- ( j = ( k + K ) -> A = B ) |
6 |
|
nfcv |
|- F/_ m A |
7 |
|
nfcsb1v |
|- F/_ j [_ m / j ]_ A |
8 |
|
csbeq1a |
|- ( j = m -> A = [_ m / j ]_ A ) |
9 |
6 7 8
|
cbvsumi |
|- sum_ j e. ( M ... N ) A = sum_ m e. ( M ... N ) [_ m / j ]_ A |
10 |
1
|
znegcld |
|- ( ph -> -u K e. ZZ ) |
11 |
4
|
ralrimiva |
|- ( ph -> A. j e. ( M ... N ) A e. CC ) |
12 |
7
|
nfel1 |
|- F/ j [_ m / j ]_ A e. CC |
13 |
8
|
eleq1d |
|- ( j = m -> ( A e. CC <-> [_ m / j ]_ A e. CC ) ) |
14 |
12 13
|
rspc |
|- ( m e. ( M ... N ) -> ( A. j e. ( M ... N ) A e. CC -> [_ m / j ]_ A e. CC ) ) |
15 |
11 14
|
mpan9 |
|- ( ( ph /\ m e. ( M ... N ) ) -> [_ m / j ]_ A e. CC ) |
16 |
|
csbeq1 |
|- ( m = ( k - -u K ) -> [_ m / j ]_ A = [_ ( k - -u K ) / j ]_ A ) |
17 |
10 2 3 15 16
|
fsumshft |
|- ( ph -> sum_ m e. ( M ... N ) [_ m / j ]_ A = sum_ k e. ( ( M + -u K ) ... ( N + -u K ) ) [_ ( k - -u K ) / j ]_ A ) |
18 |
2
|
zcnd |
|- ( ph -> M e. CC ) |
19 |
1
|
zcnd |
|- ( ph -> K e. CC ) |
20 |
18 19
|
negsubd |
|- ( ph -> ( M + -u K ) = ( M - K ) ) |
21 |
3
|
zcnd |
|- ( ph -> N e. CC ) |
22 |
21 19
|
negsubd |
|- ( ph -> ( N + -u K ) = ( N - K ) ) |
23 |
20 22
|
oveq12d |
|- ( ph -> ( ( M + -u K ) ... ( N + -u K ) ) = ( ( M - K ) ... ( N - K ) ) ) |
24 |
23
|
sumeq1d |
|- ( ph -> sum_ k e. ( ( M + -u K ) ... ( N + -u K ) ) [_ ( k - -u K ) / j ]_ A = sum_ k e. ( ( M - K ) ... ( N - K ) ) [_ ( k - -u K ) / j ]_ A ) |
25 |
|
elfzelz |
|- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. ZZ ) |
26 |
25
|
zcnd |
|- ( k e. ( ( M - K ) ... ( N - K ) ) -> k e. CC ) |
27 |
|
subneg |
|- ( ( k e. CC /\ K e. CC ) -> ( k - -u K ) = ( k + K ) ) |
28 |
26 19 27
|
syl2anr |
|- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> ( k - -u K ) = ( k + K ) ) |
29 |
28
|
csbeq1d |
|- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k - -u K ) / j ]_ A = [_ ( k + K ) / j ]_ A ) |
30 |
|
ovex |
|- ( k + K ) e. _V |
31 |
30 5
|
csbie |
|- [_ ( k + K ) / j ]_ A = B |
32 |
29 31
|
eqtrdi |
|- ( ( ph /\ k e. ( ( M - K ) ... ( N - K ) ) ) -> [_ ( k - -u K ) / j ]_ A = B ) |
33 |
32
|
sumeq2dv |
|- ( ph -> sum_ k e. ( ( M - K ) ... ( N - K ) ) [_ ( k - -u K ) / j ]_ A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B ) |
34 |
17 24 33
|
3eqtrd |
|- ( ph -> sum_ m e. ( M ... N ) [_ m / j ]_ A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B ) |
35 |
9 34
|
eqtrid |
|- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B ) |