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Theorem fsumshftm 12556
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 2571 . . 3  |-  F/_ m A
2 nfcsb1v 3275 . . 3  |-  F/_ j [_ m  /  j ]_ A
3 csbeq1a 3251 . . 3  |-  ( j  =  m  ->  A  =  [_ m  /  j ]_ A )
41, 2, 3cbvsumi 12483 . 2  |-  sum_ j  e.  ( M ... N
) A  =  sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A
5 fsumrev.1 . . . . 5  |-  ( ph  ->  K  e.  ZZ )
65znegcld 10369 . . . 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 2781 . . . . 5  |-  ( ph  ->  A. j  e.  ( M ... N ) A  e.  CC )
112nfel1 2581 . . . . . 6  |-  F/ j
[_ m  /  j ]_ A  e.  CC
123eleq1d 2501 . . . . . 6  |-  ( j  =  m  ->  ( A  e.  CC  <->  [_ m  / 
j ]_ A  e.  CC ) )
1311, 12rspc 3038 . . . . 5  |-  ( m  e.  ( M ... N )  ->  ( A. j  e.  ( M ... N ) A  e.  CC  ->  [_ m  /  j ]_ A  e.  CC ) )
1410, 13mpan9 456 . . . 4  |-  ( (
ph  /\  m  e.  ( M ... N ) )  ->  [_ m  / 
j ]_ A  e.  CC )
15 csbeq1 3246 . . . 4  |-  ( m  =  ( k  -  -u K )  ->  [_ m  /  j ]_ A  =  [_ ( k  -  -u K )  /  j ]_ A )
166, 7, 8, 14, 15fsumshft 12555 . . 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 10368 . . . . . 6  |-  ( ph  ->  M  e.  CC )
185zcnd 10368 . . . . . 6  |-  ( ph  ->  K  e.  CC )
1917, 18negsubd 9409 . . . . 5  |-  ( ph  ->  ( M  +  -u K )  =  ( M  -  K ) )
208zcnd 10368 . . . . . 6  |-  ( ph  ->  N  e.  CC )
2120, 18negsubd 9409 . . . . 5  |-  ( ph  ->  ( N  +  -u K )  =  ( N  -  K ) )
2219, 21oveq12d 6091 . . . 4  |-  ( ph  ->  ( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K ) ) )
2322sumeq1d 12487 . . 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 11051 . . . . . . . 8  |-  ( k  e.  ( ( M  -  K ) ... ( N  -  K
) )  ->  k  e.  ZZ )
2524zcnd 10368 . . . . . . 7  |-  ( k  e.  ( ( M  -  K ) ... ( N  -  K
) )  ->  k  e.  CC )
26 subneg 9342 . . . . . . 7  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( k  -  -u K
)  =  ( k  +  K ) )
2725, 18, 26syl2anr 465 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  (
k  -  -u K
)  =  ( k  +  K ) )
2827csbeq1d 3249 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  [_ (
k  -  -u K
)  /  j ]_ A  =  [_ ( k  +  K )  / 
j ]_ A )
29 ovex 6098 . . . . . 6  |-  ( k  +  K )  e. 
_V
30 nfcv 2571 . . . . . 6  |-  F/_ j B
31 fsumshftm.5 . . . . . 6  |-  ( j  =  ( k  +  K )  ->  A  =  B )
3229, 30, 31csbief 3284 . . . . 5  |-  [_ (
k  +  K )  /  j ]_ A  =  B
3328, 32syl6eq 2483 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  [_ (
k  -  -u K
)  /  j ]_ A  =  B )
3433sumeq2dv 12489 . . 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 2471 . 2  |-  ( ph  -> 
sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A  =  sum_ k  e.  ( ( M  -  K
) ... ( N  -  K ) ) B )
364, 35syl5eq 2479 1  |-  ( ph  -> 
sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  -  K ) ... ( N  -  K )
) B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   [_csb 3243  (class class class)co 6073   CCcc 8980    + caddc 8985    - cmin 9283   -ucneg 9284   ZZcz 10274   ...cfz 11035   sum_csu 12471
This theorem is referenced by:  fsumtscopo  12573  fsumparts  12577  arisum  12631  geo2sum  12642  ovolicc2lem4  19408  uniioombllem3  19469  dvply1  20193  pserdvlem2  20336  advlogexp  20538  dchrisumlem1  21175  pntpbnd2  21273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472
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