Metamath Proof Explorer


Theorem fsumshftm

Description: Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005) (Revised by Mario Carneiro, 24-Apr-2014)

Ref Expression
Hypotheses fsumrev.1
|- ( ph -> K e. ZZ )
fsumrev.2
|- ( ph -> M e. ZZ )
fsumrev.3
|- ( ph -> N e. ZZ )
fsumrev.4
|- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
fsumshftm.5
|- ( j = ( k + K ) -> A = B )
Assertion fsumshftm
|- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )

Proof

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 )