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Theorem fsumshftm 12243
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 2419 . . 3  |-  F/_ m A
2 nfcsb1v 3113 . . 3  |-  F/_ j [_ m  /  j ]_ A
3 csbeq1a 3089 . . 3  |-  ( j  =  m  ->  A  =  [_ m  /  j ]_ A )
41, 2, 3cbvsumi 12170 . 2  |-  sum_ j  e.  ( M ... N
) A  =  sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A
5 fsumrev.1 . . . . 5  |-  ( ph  ->  K  e.  ZZ )
65znegcld 10119 . . . 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 2626 . . . . 5  |-  ( ph  ->  A. j  e.  ( M ... N ) A  e.  CC )
112nfel1 2429 . . . . . 6  |-  F/ j
[_ m  /  j ]_ A  e.  CC
123eleq1d 2349 . . . . . 6  |-  ( j  =  m  ->  ( A  e.  CC  <->  [_ m  / 
j ]_ A  e.  CC ) )
1311, 12rspc 2878 . . . . 5  |-  ( m  e.  ( M ... N )  ->  ( A. j  e.  ( M ... N ) A  e.  CC  ->  [_ m  /  j ]_ A  e.  CC ) )
1410, 13mpan9 455 . . . 4  |-  ( (
ph  /\  m  e.  ( M ... N ) )  ->  [_ m  / 
j ]_ A  e.  CC )
15 csbeq1 3084 . . . 4  |-  ( m  =  ( k  -  -u K )  ->  [_ m  /  j ]_ A  =  [_ ( k  -  -u K )  /  j ]_ A )
166, 7, 8, 14, 15fsumshft 12242 . . 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 10118 . . . . . 6  |-  ( ph  ->  M  e.  CC )
185zcnd 10118 . . . . . 6  |-  ( ph  ->  K  e.  CC )
1917, 18negsubd 9163 . . . . 5  |-  ( ph  ->  ( M  +  -u K )  =  ( M  -  K ) )
208zcnd 10118 . . . . . 6  |-  ( ph  ->  N  e.  CC )
2120, 18negsubd 9163 . . . . 5  |-  ( ph  ->  ( N  +  -u K )  =  ( N  -  K ) )
2219, 21oveq12d 5876 . . . 4  |-  ( ph  ->  ( ( M  +  -u K ) ... ( N  +  -u K ) )  =  ( ( M  -  K ) ... ( N  -  K ) ) )
2322sumeq1d 12174 . . 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 10798 . . . . . . . 8  |-  ( k  e.  ( ( M  -  K ) ... ( N  -  K
) )  ->  k  e.  ZZ )
2524zcnd 10118 . . . . . . 7  |-  ( k  e.  ( ( M  -  K ) ... ( N  -  K
) )  ->  k  e.  CC )
26 subneg 9096 . . . . . . 7  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( k  -  -u K
)  =  ( k  +  K ) )
2725, 18, 26syl2anr 464 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  (
k  -  -u K
)  =  ( k  +  K ) )
2827csbeq1d 3087 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  [_ (
k  -  -u K
)  /  j ]_ A  =  [_ ( k  +  K )  / 
j ]_ A )
29 ovex 5883 . . . . . 6  |-  ( k  +  K )  e. 
_V
30 nfcv 2419 . . . . . 6  |-  F/_ j B
31 fsumshftm.5 . . . . . 6  |-  ( j  =  ( k  +  K )  ->  A  =  B )
3229, 30, 31csbief 3122 . . . . 5  |-  [_ (
k  +  K )  /  j ]_ A  =  B
3328, 32syl6eq 2331 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M  -  K ) ... ( N  -  K )
) )  ->  [_ (
k  -  -u K
)  /  j ]_ A  =  B )
3433sumeq2dv 12176 . . 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 2319 . 2  |-  ( ph  -> 
sum_ m  e.  ( M ... N ) [_ m  /  j ]_ A  =  sum_ k  e.  ( ( M  -  K
) ... ( N  -  K ) ) B )
364, 35syl5eq 2327 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   [_csb 3081  (class class class)co 5858   CCcc 8735    + caddc 8740    - cmin 9037   -ucneg 9038   ZZcz 10024   ...cfz 10782   sum_csu 12158
This theorem is referenced by:  fsumtscopo  12260  fsumparts  12264  arisum  12318  geo2sum  12329  ovolicc2lem4  18879  uniioombllem3  18940  dvply1  19664  pserdvlem2  19804  advlogexp  20002  dchrisumlem1  20638  pntpbnd2  20736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159
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