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Theorem fsumshft 12518
Description: Index shift of a finite sum. (Contributed by NM, 27-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 )
fsumshft.5  |-  ( j  =  ( k  -  K )  ->  A  =  B )
Assertion
Ref Expression
fsumshft  |-  ( 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 fsumshft
StepHypRef Expression
1 fsumshft.5 . 2  |-  ( j  =  ( k  -  K )  ->  A  =  B )
2 fzfid 11267 . 2  |-  ( ph  ->  ( ( M  +  K ) ... ( N  +  K )
)  e.  Fin )
3 ovex 6065 . . . . 5  |-  ( j  -  K )  e. 
_V
4 eqid 2404 . . . . 5  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )
53, 4fnmpti 5532 . . . 4  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  Fn  ( ( M  +  K ) ... ( N  +  K
) )
65a1i 11 . . 3  |-  ( ph  ->  ( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) )  Fn  ( ( M  +  K ) ... ( N  +  K ) ) )
7 ovex 6065 . . . . 5  |-  ( k  +  K )  e. 
_V
8 eqid 2404 . . . . 5  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  =  ( k  e.  ( M ... N
)  |->  ( k  +  K ) )
97, 8fnmpti 5532 . . . 4  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  Fn  ( M ... N )
10 simprr 734 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  =  ( j  -  K ) )
1110oveq1d 6055 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  =  ( ( j  -  K )  +  K ) )
12 elfzelz 11015 . . . . . . . . . . . . . 14  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  j  e.  ZZ )
1312ad2antrl 709 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ZZ )
14 fsumrev.1 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  ZZ )
1514adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  K  e.  ZZ )
16 zcn 10243 . . . . . . . . . . . . . 14  |-  ( j  e.  ZZ  ->  j  e.  CC )
17 zcn 10243 . . . . . . . . . . . . . 14  |-  ( K  e.  ZZ  ->  K  e.  CC )
18 npcan 9270 . . . . . . . . . . . . . 14  |-  ( ( j  e.  CC  /\  K  e.  CC )  ->  ( ( j  -  K )  +  K
)  =  j )
1916, 17, 18syl2an 464 . . . . . . . . . . . . 13  |-  ( ( j  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( j  -  K )  +  K
)  =  j )
2013, 15, 19syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( ( j  -  K )  +  K
)  =  j )
2111, 20eqtr2d 2437 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  =  ( k  +  K ) )
22 simprl 733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
2321, 22eqeltrrd 2479 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
24 fsumrev.2 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ZZ )
2524adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  M  e.  ZZ )
26 fsumrev.3 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  ZZ )
2726adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  N  e.  ZZ )
2813, 15zsubcld 10336 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( j  -  K
)  e.  ZZ )
2910, 28eqeltrd 2478 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ZZ )
30 fzaddel 11043 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
3125, 27, 29, 15, 30syl22anc 1185 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
3223, 31mpbird 224 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ( M ... N ) )
3332, 21jca 519 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( k  +  K ) ) )
34 simprr 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  =  ( k  +  K ) )
35 simprl 733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ( M ... N ) )
3624adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  M  e.  ZZ )
3726adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  N  e.  ZZ )
38 elfzelz 11015 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
3938ad2antrl 709 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ZZ )
4014adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  K  e.  ZZ )
4136, 37, 39, 40, 30syl22anc 1185 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
4235, 41mpbid 202 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
4334, 42eqeltrd 2478 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
4434oveq1d 6055 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  -  K
)  =  ( ( k  +  K )  -  K ) )
45 zcn 10243 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  k  e.  CC )
46 pncan 9267 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( ( k  +  K )  -  K
)  =  k )
4745, 17, 46syl2an 464 . . . . . . . . . . 11  |-  ( ( k  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( k  +  K )  -  K
)  =  k )
4839, 40, 47syl2anc 643 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( ( k  +  K )  -  K
)  =  k )
4944, 48eqtr2d 2437 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  =  ( j  -  K ) )
5043, 49jca 519 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) )
5133, 50impbida 806 . . . . . . 7  |-  ( ph  ->  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) ) )
5251opabbidv 4231 . . . . . 6  |-  ( ph  ->  { <. k ,  j
>.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) }  =  { <. k ,  j
>.  |  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) } )
53 df-mpt 4228 . . . . . . . 8  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  { <. j ,  k >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) }
5453cnveqi 5006 . . . . . . 7  |-  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  =  `' { <. j ,  k >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) }
55 cnvopab 5233 . . . . . . 7  |-  `' { <. j ,  k >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) }  =  { <. k ,  j
>.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) }
5654, 55eqtri 2424 . . . . . 6  |-  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  =  { <. k ,  j >.  |  ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) }
57 df-mpt 4228 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  =  { <. k ,  j >.  |  ( k  e.  ( M ... N )  /\  j  =  ( k  +  K ) ) }
5852, 56, 573eqtr4g 2461 . . . . 5  |-  ( ph  ->  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  =  ( k  e.  ( M ... N ) 
|->  ( k  +  K
) ) )
5958fneq1d 5495 . . . 4  |-  ( ph  ->  ( `' ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  Fn  ( M ... N )  <->  ( k  e.  ( M ... N
)  |->  ( k  +  K ) )  Fn  ( M ... N
) ) )
609, 59mpbiri 225 . . 3  |-  ( ph  ->  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  Fn  ( M ... N
) )
61 dff1o4 5641 . . 3  |-  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) ) : ( ( M  +  K ) ... ( N  +  K ) ) -1-1-onto-> ( M ... N )  <->  ( (
j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  Fn  ( ( M  +  K ) ... ( N  +  K ) )  /\  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  Fn  ( M ... N
) ) )
626, 60, 61sylanbrc 646 . 2  |-  ( ph  ->  ( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) : ( ( M  +  K ) ... ( N  +  K ) ) -1-1-onto-> ( M ... N ) )
63 oveq1 6047 . . . 4  |-  ( j  =  k  ->  (
j  -  K )  =  ( k  -  K ) )
64 ovex 6065 . . . 4  |-  ( k  -  K )  e. 
_V
6563, 4, 64fvmpt 5765 . . 3  |-  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  (
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) `  k )  =  ( k  -  K ) )
6665adantl 453 . 2  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) `  k )  =  ( k  -  K ) )
67 fsumrev.4 . 2  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
681, 2, 62, 66, 67fsumf1o 12472 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    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {copab 4225    e. cmpt 4226   `'ccnv 4836    Fn wfn 5408   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944    + caddc 8949    - cmin 9247   ZZcz 10238   ...cfz 10999   sum_csu 12434
This theorem is referenced by:  fsumshftm  12519  binomlem  12563  dvtaylp  20239  binomfallfaclem2  25307  bpolydiflem  26004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435
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