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Theorem isumsplit 12299
Description: Split off the first  N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
isumsplit.1  |-  Z  =  ( ZZ>= `  M )
isumsplit.2  |-  W  =  ( ZZ>= `  N )
isumsplit.3  |-  ( ph  ->  N  e.  Z )
isumsplit.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
isumsplit.5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
isumsplit.6  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
Assertion
Ref Expression
isumsplit  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
Distinct variable groups:    k, F    k, M    ph, k    k, Z   
k, N    k, W
Allowed substitution hint:    A( k)

Proof of Theorem isumsplit
Dummy variables  j  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isumsplit.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 isumsplit.3 . . . 4  |-  ( ph  ->  N  e.  Z )
32, 1syl6eleq 2373 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 eluzel2 10235 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
53, 4syl 15 . 2  |-  ( ph  ->  M  e.  ZZ )
6 isumsplit.4 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
7 isumsplit.5 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
8 isumsplit.2 . . 3  |-  W  =  ( ZZ>= `  N )
9 eluzelz 10238 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
103, 9syl 15 . . 3  |-  ( ph  ->  N  e.  ZZ )
11 uzss 10248 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
123, 11syl 15 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
1312, 8, 13sstr4g 3219 . . . . . 6  |-  ( ph  ->  W  C_  Z )
1413sselda 3180 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  k  e.  Z )
1514, 6syldan 456 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  =  A )
1614, 7syldan 456 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  CC )
17 isumsplit.6 . . . . 5  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
186, 7eqeltrd 2357 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
191, 2, 18iserex 12130 . . . . 5  |-  ( ph  ->  (  seq  M (  +  ,  F )  e.  dom  ~~>  <->  seq  N (  +  ,  F )  e.  dom  ~~>  ) )
2017, 19mpbid 201 . . . 4  |-  ( ph  ->  seq  N (  +  ,  F )  e. 
dom 
~~>  )
218, 10, 15, 16, 20isumclim2 12221 . . 3  |-  ( ph  ->  seq  N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
22 fzfid 11035 . . . 4  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
23 elfzuz 10794 . . . . . 6  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
2423, 1syl6eleqr 2374 . . . . 5  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
2524, 7sylan2 460 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
2622, 25fsumcl 12206 . . 3  |-  ( ph  -> 
sum_ k  e.  ( M ... ( N  -  1 ) ) A  e.  CC )
2714, 18syldan 456 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  e.  CC )
288, 10, 27serf 11074 . . . 4  |-  ( ph  ->  seq  N (  +  ,  F ) : W --> CC )
29 ffvelrn 5663 . . . 4  |-  ( (  seq  N (  +  ,  F ) : W --> CC  /\  j  e.  W )  ->  (  seq  N (  +  ,  F ) `  j
)  e.  CC )
3028, 29sylan 457 . . 3  |-  ( (
ph  /\  j  e.  W )  ->  (  seq  N (  +  ,  F ) `  j
)  e.  CC )
315zred 10117 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  RR )
3231ltm1d 9689 . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  1 )  <  M )
33 peano2zm 10062 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
345, 33syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
35 fzn 10810 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
365, 34, 35syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( ( M  - 
1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
3732, 36mpbid 201 . . . . . . . . . 10  |-  ( ph  ->  ( M ... ( M  -  1 ) )  =  (/) )
3837sumeq1d 12174 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  sum_ k  e.  (/)  A )
3938adantr 451 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  sum_ k  e.  (/)  A )
40 sum0 12194 . . . . . . . 8  |-  sum_ k  e.  (/)  A  =  0
4139, 40syl6eq 2331 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  0 )
4241oveq1d 5873 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  ( sum_ k  e.  ( M ... ( M  - 
1 ) ) A  +  (  seq  M
(  +  ,  F
) `  j )
)  =  ( 0  +  (  seq  M
(  +  ,  F
) `  j )
) )
4313sselda 3180 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  j  e.  Z )
441, 5, 18serf 11074 . . . . . . . . 9  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
45 ffvelrn 5663 . . . . . . . . 9  |-  ( (  seq  M (  +  ,  F ) : Z --> CC  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  CC )
4644, 45sylan 457 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  CC )
4743, 46syldan 456 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  CC )
4847addid2d 9013 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  (
0  +  (  seq 
M (  +  ,  F ) `  j
) )  =  (  seq  M (  +  ,  F ) `  j ) )
4942, 48eqtr2d 2316 . . . . 5  |-  ( (
ph  /\  j  e.  W )  ->  (  seq  M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( M  -  1 ) ) A  +  (  seq  M (  +  ,  F ) `  j ) ) )
50 oveq1 5865 . . . . . . . . 9  |-  ( N  =  M  ->  ( N  -  1 )  =  ( M  - 
1 ) )
5150oveq2d 5874 . . . . . . . 8  |-  ( N  =  M  ->  ( M ... ( N  - 
1 ) )  =  ( M ... ( M  -  1 ) ) )
5251sumeq1d 12174 . . . . . . 7  |-  ( N  =  M  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  sum_ k  e.  ( M ... ( M  -  1 ) ) A )
53 seqeq1 11049 . . . . . . . 8  |-  ( N  =  M  ->  seq  N (  +  ,  F
)  =  seq  M
(  +  ,  F
) )
5453fveq1d 5527 . . . . . . 7  |-  ( N  =  M  ->  (  seq  N (  +  ,  F ) `  j
)  =  (  seq 
M (  +  ,  F ) `  j
) )
5552, 54oveq12d 5876 . . . . . 6  |-  ( N  =  M  ->  ( sum_ k  e.  ( M ... ( N  - 
1 ) ) A  +  (  seq  N
(  +  ,  F
) `  j )
)  =  ( sum_ k  e.  ( M ... ( M  -  1 ) ) A  +  (  seq  M (  +  ,  F ) `  j ) ) )
5655eqeq2d 2294 . . . . 5  |-  ( N  =  M  ->  (
(  seq  M (  +  ,  F ) `  j )  =  (
sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq 
N (  +  ,  F ) `  j
) )  <->  (  seq  M (  +  ,  F
) `  j )  =  ( sum_ k  e.  ( M ... ( M  -  1 ) ) A  +  (  seq  M (  +  ,  F ) `  j ) ) ) )
5749, 56syl5ibrcom 213 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  =  M  ->  (  seq  M (  +  ,  F ) `  j )  =  (
sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq 
N (  +  ,  F ) `  j
) ) ) )
58 addcl 8819 . . . . . . . 8  |-  ( ( k  e.  CC  /\  m  e.  CC )  ->  ( k  +  m
)  e.  CC )
5958adantl 452 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  ( k  e.  CC  /\  m  e.  CC ) )  -> 
( k  +  m
)  e.  CC )
60 addass 8824 . . . . . . . 8  |-  ( ( k  e.  CC  /\  m  e.  CC  /\  x  e.  CC )  ->  (
( k  +  m
)  +  x )  =  ( k  +  ( m  +  x
) ) )
6160adantl 452 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  ( k  e.  CC  /\  m  e.  CC  /\  x  e.  CC ) )  -> 
( ( k  +  m )  +  x
)  =  ( k  +  ( m  +  x ) ) )
62 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  j  e.  W )
63 simpll 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ph )
6410zcnd 10118 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
65 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
66 npcan 9060 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
6764, 65, 66sylancl 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
6867eqcomd 2288 . . . . . . . . . . 11  |-  ( ph  ->  N  =  ( ( N  -  1 )  +  1 ) )
6963, 68syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  N  =  ( ( N  - 
1 )  +  1 ) )
7069fveq2d 5529 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( ZZ>= `  N )  =  (
ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
718, 70syl5eq 2327 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  W  =  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
7262, 71eleqtrd 2359 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  j  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
735adantr 451 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  M  e.  ZZ )
74 eluzp1m1 10251 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  M ) )
7573, 74sylan 457 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
76 elfzuz 10794 . . . . . . . . 9  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
7776, 1syl6eleqr 2374 . . . . . . . 8  |-  ( k  e.  ( M ... j )  ->  k  e.  Z )
7863, 77, 18syl2an 463 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... j ) )  ->  ( F `  k )  e.  CC )
7959, 61, 72, 75, 78seqsplit 11079 . . . . . 6  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  j )  =  ( (  seq 
M (  +  ,  F ) `  ( N  -  1 ) )  +  (  seq  ( ( N  - 
1 )  +  1 ) (  +  ,  F ) `  j
) ) )
8063, 24, 6syl2an 463 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  k )  =  A )
8163, 24, 7syl2an 463 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
8280, 75, 81fsumser 12203 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  (  seq  M (  +  ,  F ) `  ( N  -  1
) ) )
8369seqeq1d 11052 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  seq  N (  +  ,  F )  =  seq  ( ( N  -  1 )  +  1 ) (  +  ,  F ) )
8483fveq1d 5527 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  N (  +  ,  F
) `  j )  =  (  seq  ( ( N  -  1 )  +  1 ) (  +  ,  F ) `
 j ) )
8582, 84oveq12d 5876 . . . . . 6  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq  N (  +  ,  F ) `  j ) )  =  ( (  seq  M
(  +  ,  F
) `  ( N  -  1 ) )  +  (  seq  (
( N  -  1 )  +  1 ) (  +  ,  F
) `  j )
) )
8679, 85eqtr4d 2318 . . . . 5  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  j )  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq  N (  +  ,  F ) `  j ) ) )
8786ex 423 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  e.  ( ZZ>= `  ( M  +  1
) )  ->  (  seq  M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq  N (  +  ,  F ) `  j ) ) ) )
88 uzp1 10261 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
893, 88syl 15 . . . . 5  |-  ( ph  ->  ( N  =  M  \/  N  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
9089adantr 451 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1
) ) ) )
9157, 87, 90mpjaod 370 . . 3  |-  ( (
ph  /\  j  e.  W )  ->  (  seq  M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq  N (  +  ,  F ) `  j ) ) )
928, 10, 21, 26, 17, 30, 91climaddc2 12109 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
931, 5, 6, 7, 92isumclim 12220 1  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   class class class wbr 4023   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046    ~~> cli 11958   sum_csu 12158
This theorem is referenced by:  isum1p  12300  geolim2  12327  mertenslem2  12341  mertens  12342  effsumlt  12391  eirrlem  12482  rpnnen2lem8  12500  prmreclem6  12968  aaliou3lem7  19729  abelthlem7  19814  log2tlbnd  20241  subfaclim  23719  stirlinglem12  27834
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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