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Theorem fsumshftm 13244
Description: Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
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
Ref Expression
fsumshftm  |-  ( ph  -> 
sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  -  K ) ... ( N  -  K )
) B )
Distinct variable groups:    A, k    B, j    j, k, K   
j, M, k    j, N, k    ph, j, k
Allowed substitution hints:    A( j)    B( k)

Proof of Theorem fsumshftm
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 nfcv 2577 . . 3  |-  F/_ m A
2 nfcsb1v 3301 . . 3  |-  F/_ j [_ m  /  j ]_ A
3 csbeq1a 3294 . . 3  |-  ( j  =  m  ->  A  =  [_ m  /  j ]_ A )
41, 2, 3cbvsumi 13170 . 2  |-  sum_ j  e.  ( M ... N
) A  =  sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A
5 fsumrev.1 . . . . 5  |-  ( ph  ->  K  e.  ZZ )
65znegcld 10745 . . . 4  |-  ( ph  -> 
-u K  e.  ZZ )
7 fsumrev.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
8 fsumrev.3 . . . 4  |-  ( ph  ->  N  e.  ZZ )
9 fsumrev.4 . . . . . 6  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
109ralrimiva 2797 . . . . 5  |-  ( ph  ->  A. j  e.  ( M ... N ) A  e.  CC )
112nfel1 2587 . . . . . 6  |-  F/ j
[_ m  /  j ]_ A  e.  CC
123eleq1d 2507 . . . . . 6  |-  ( j  =  m  ->  ( A  e.  CC  <->  [_ m  / 
j ]_ A  e.  CC ) )
1311, 12rspc 3064 . . . . 5  |-  ( m  e.  ( M ... N )  ->  ( A. j  e.  ( M ... N ) A  e.  CC  ->  [_ m  /  j ]_ A  e.  CC ) )
1410, 13mpan9 466 . . . 4  |-  ( (
ph  /\  m  e.  ( M ... N ) )  ->  [_ m  / 
j ]_ A  e.  CC )
15 csbeq1 3288 . . . 4  |-  ( m  =  ( k  -  -u K )  ->  [_ m  /  j ]_ A  =  [_ ( k  -  -u K )  /  j ]_ A )
166, 7, 8, 14, 15fsumshft 13243 . . 3  |-  ( ph  -> 
sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A  =  sum_ k  e.  ( ( M  +  -u K ) ... ( N  +  -u K ) ) [_ ( k  -  -u K )  / 
j ]_ A )
177zcnd 10744 . . . . . 6  |-  ( ph  ->  M  e.  CC )
185zcnd 10744 . . . . . 6  |-  ( ph  ->  K  e.  CC )
1917, 18negsubd 9721 . . . . 5  |-  ( ph  ->  ( M  +  -u K )  =  ( M  -  K ) )
208zcnd 10744 . . . . . 6  |-  ( ph  ->  N  e.  CC )
2120, 18negsubd 9721 . . . . 5  |-  ( ph  ->  ( N  +  -u K )  =  ( N  -  K ) )
2219, 21oveq12d 6108 . . . 4  |-  ( ph  ->  ( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K ) ) )
2322sumeq1d 13174 . . 3  |-  ( 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 )
24 elfzelz 11449 . . . . . . . 8  |-  ( k  e.  ( ( M  -  K ) ... ( N  -  K
) )  ->  k  e.  ZZ )
2524zcnd 10744 . . . . . . 7  |-  ( k  e.  ( ( M  -  K ) ... ( N  -  K
) )  ->  k  e.  CC )
26 subneg 9654 . . . . . . 7  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( k  -  -u K
)  =  ( k  +  K ) )
2725, 18, 26syl2anr 475 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  (
k  -  -u K
)  =  ( k  +  K ) )
2827csbeq1d 3292 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  [_ (
k  -  -u K
)  /  j ]_ A  =  [_ ( k  +  K )  / 
j ]_ A )
29 ovex 6115 . . . . . 6  |-  ( k  +  K )  e. 
_V
30 nfcv 2577 . . . . . 6  |-  F/_ j B
31 fsumshftm.5 . . . . . 6  |-  ( j  =  ( k  +  K )  ->  A  =  B )
3229, 30, 31csbief 3310 . . . . 5  |-  [_ (
k  +  K )  /  j ]_ A  =  B
3328, 32syl6eq 2489 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  [_ (
k  -  -u K
)  /  j ]_ A  =  B )
3433sumeq2dv 13176 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ( M  -  K
) ... ( N  -  K ) ) [_ ( k  -  -u K
)  /  j ]_ A  =  sum_ k  e.  ( ( M  -  K ) ... ( N  -  K )
) B )
3516, 23, 343eqtrd 2477 . 2  |-  ( ph  -> 
sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A  =  sum_ k  e.  ( ( M  -  K
) ... ( N  -  K ) ) B )
364, 35syl5eq 2485 1  |-  ( ph  -> 
sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  -  K ) ... ( N  -  K )
) B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   [_csb 3285  (class class class)co 6090   CCcc 9276    + caddc 9281    - cmin 9591   -ucneg 9592   ZZcz 10642   ...cfz 11433   sum_csu 13159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160
This theorem is referenced by:  fsumtscopo  13261  fsumparts  13265  arisum  13318  geo2sum  13329  ovolicc2lem4  20903  uniioombllem3  20965  dvply1  21693  pserdvlem2  21836  advlogexp  22043  dchrisumlem1  22681  pntpbnd2  22779
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