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Theorem fsumshftm 13745
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 2564 . . 3  |-  F/_ m A
2 nfcsb1v 3388 . . 3  |-  F/_ j [_ m  /  j ]_ A
3 csbeq1a 3381 . . 3  |-  ( j  =  m  ->  A  =  [_ m  /  j ]_ A )
41, 2, 3cbvsumi 13666 . 2  |-  sum_ j  e.  ( M ... N
) A  =  sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A
5 fsumrev.1 . . . . 5  |-  ( ph  ->  K  e.  ZZ )
65znegcld 11009 . . . 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 2817 . . . . 5  |-  ( ph  ->  A. j  e.  ( M ... N ) A  e.  CC )
112nfel1 2580 . . . . . 6  |-  F/ j
[_ m  /  j ]_ A  e.  CC
123eleq1d 2471 . . . . . 6  |-  ( j  =  m  ->  ( A  e.  CC  <->  [_ m  / 
j ]_ A  e.  CC ) )
1311, 12rspc 3153 . . . . 5  |-  ( m  e.  ( M ... N )  ->  ( A. j  e.  ( M ... N ) A  e.  CC  ->  [_ m  /  j ]_ A  e.  CC ) )
1410, 13mpan9 467 . . . 4  |-  ( (
ph  /\  m  e.  ( M ... N ) )  ->  [_ m  / 
j ]_ A  e.  CC )
15 csbeq1 3375 . . . 4  |-  ( m  =  ( k  -  -u K )  ->  [_ m  /  j ]_ A  =  [_ ( k  -  -u K )  /  j ]_ A )
166, 7, 8, 14, 15fsumshft 13744 . . 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 11008 . . . . . 6  |-  ( ph  ->  M  e.  CC )
185zcnd 11008 . . . . . 6  |-  ( ph  ->  K  e.  CC )
1917, 18negsubd 9972 . . . . 5  |-  ( ph  ->  ( M  +  -u K )  =  ( M  -  K ) )
208zcnd 11008 . . . . . 6  |-  ( ph  ->  N  e.  CC )
2120, 18negsubd 9972 . . . . 5  |-  ( ph  ->  ( N  +  -u K )  =  ( N  -  K ) )
2219, 21oveq12d 6295 . . . 4  |-  ( ph  ->  ( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K ) ) )
2322sumeq1d 13670 . . 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 11740 . . . . . . . 8  |-  ( k  e.  ( ( M  -  K ) ... ( N  -  K
) )  ->  k  e.  ZZ )
2524zcnd 11008 . . . . . . 7  |-  ( k  e.  ( ( M  -  K ) ... ( N  -  K
) )  ->  k  e.  CC )
26 subneg 9903 . . . . . . 7  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( k  -  -u K
)  =  ( k  +  K ) )
2725, 18, 26syl2anr 476 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  (
k  -  -u K
)  =  ( k  +  K ) )
2827csbeq1d 3379 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  [_ (
k  -  -u K
)  /  j ]_ A  =  [_ ( k  +  K )  / 
j ]_ A )
29 ovex 6305 . . . . . 6  |-  ( k  +  K )  e. 
_V
30 fsumshftm.5 . . . . . 6  |-  ( j  =  ( k  +  K )  ->  A  =  B )
3129, 30csbie 3398 . . . . 5  |-  [_ (
k  +  K )  /  j ]_ A  =  B
3228, 31syl6eq 2459 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  [_ (
k  -  -u K
)  /  j ]_ A  =  B )
3332sumeq2dv 13672 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ( M  -  K
) ... ( N  -  K ) ) [_ ( k  -  -u K
)  /  j ]_ A  =  sum_ k  e.  ( ( M  -  K ) ... ( N  -  K )
) B )
3416, 23, 333eqtrd 2447 . 2  |-  ( ph  -> 
sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A  =  sum_ k  e.  ( ( M  -  K
) ... ( N  -  K ) ) B )
354, 34syl5eq 2455 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 367    = wceq 1405    e. wcel 1842   A.wral 2753   [_csb 3372  (class class class)co 6277   CCcc 9519    + caddc 9524    - cmin 9840   -ucneg 9841   ZZcz 10904   ...cfz 11724   sum_csu 13655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-fz 11725  df-fzo 11853  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-sum 13656
This theorem is referenced by:  telfsumo  13765  fsumparts  13769  arisum  13821  geo2sum  13832  ovolicc2lem4  22221  uniioombllem3  22284  dvply1  22970  pserdvlem2  23113  advlogexp  23328  dchrisumlem1  24053  pntpbnd2  24151  nn0sumshdiglemA  38731  nn0sumshdiglemB  38732
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