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Theorem fsumshftm 13370
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 2616 . . 3  |-  F/_ m A
2 nfcsb1v 3414 . . 3  |-  F/_ j [_ m  /  j ]_ A
3 csbeq1a 3407 . . 3  |-  ( j  =  m  ->  A  =  [_ m  /  j ]_ A )
41, 2, 3cbvsumi 13296 . 2  |-  sum_ j  e.  ( M ... N
) A  =  sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A
5 fsumrev.1 . . . . 5  |-  ( ph  ->  K  e.  ZZ )
65znegcld 10864 . . . 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 2830 . . . . 5  |-  ( ph  ->  A. j  e.  ( M ... N ) A  e.  CC )
112nfel1 2632 . . . . . 6  |-  F/ j
[_ m  /  j ]_ A  e.  CC
123eleq1d 2523 . . . . . 6  |-  ( j  =  m  ->  ( A  e.  CC  <->  [_ m  / 
j ]_ A  e.  CC ) )
1311, 12rspc 3173 . . . . 5  |-  ( m  e.  ( M ... N )  ->  ( A. j  e.  ( M ... N ) A  e.  CC  ->  [_ m  /  j ]_ A  e.  CC ) )
1410, 13mpan9 469 . . . 4  |-  ( (
ph  /\  m  e.  ( M ... N ) )  ->  [_ m  / 
j ]_ A  e.  CC )
15 csbeq1 3401 . . . 4  |-  ( m  =  ( k  -  -u K )  ->  [_ m  /  j ]_ A  =  [_ ( k  -  -u K )  /  j ]_ A )
166, 7, 8, 14, 15fsumshft 13369 . . 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 10863 . . . . . 6  |-  ( ph  ->  M  e.  CC )
185zcnd 10863 . . . . . 6  |-  ( ph  ->  K  e.  CC )
1917, 18negsubd 9840 . . . . 5  |-  ( ph  ->  ( M  +  -u K )  =  ( M  -  K ) )
208zcnd 10863 . . . . . 6  |-  ( ph  ->  N  e.  CC )
2120, 18negsubd 9840 . . . . 5  |-  ( ph  ->  ( N  +  -u K )  =  ( N  -  K ) )
2219, 21oveq12d 6221 . . . 4  |-  ( ph  ->  ( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K ) ) )
2322sumeq1d 13300 . . 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 11574 . . . . . . . 8  |-  ( k  e.  ( ( M  -  K ) ... ( N  -  K
) )  ->  k  e.  ZZ )
2524zcnd 10863 . . . . . . 7  |-  ( k  e.  ( ( M  -  K ) ... ( N  -  K
) )  ->  k  e.  CC )
26 subneg 9773 . . . . . . 7  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( k  -  -u K
)  =  ( k  +  K ) )
2725, 18, 26syl2anr 478 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  (
k  -  -u K
)  =  ( k  +  K ) )
2827csbeq1d 3405 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  [_ (
k  -  -u K
)  /  j ]_ A  =  [_ ( k  +  K )  / 
j ]_ A )
29 ovex 6228 . . . . . 6  |-  ( k  +  K )  e. 
_V
30 nfcv 2616 . . . . . 6  |-  F/_ j B
31 fsumshftm.5 . . . . . 6  |-  ( j  =  ( k  +  K )  ->  A  =  B )
3229, 30, 31csbief 3423 . . . . 5  |-  [_ (
k  +  K )  /  j ]_ A  =  B
3328, 32syl6eq 2511 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  [_ (
k  -  -u K
)  /  j ]_ A  =  B )
3433sumeq2dv 13302 . . 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 2499 . 2  |-  ( ph  -> 
sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A  =  sum_ k  e.  ( ( M  -  K
) ... ( N  -  K ) ) B )
364, 35syl5eq 2507 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 1370    e. wcel 1758   A.wral 2799   [_csb 3398  (class class class)co 6203   CCcc 9395    + caddc 9400    - cmin 9710   -ucneg 9711   ZZcz 10761   ...cfz 11558   sum_csu 13285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-fz 11559  df-fzo 11670  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-sum 13286
This theorem is referenced by:  fsumtscopo  13387  fsumparts  13391  arisum  13444  geo2sum  13455  ovolicc2lem4  21145  uniioombllem3  21208  dvply1  21893  pserdvlem2  22036  advlogexp  22243  dchrisumlem1  22881  pntpbnd2  22979
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