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Theorem isumsplit 12612
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 2525 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 eluzel2 10485 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
53, 4syl 16 . 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 10488 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
103, 9syl 16 . . 3  |-  ( ph  ->  N  e.  ZZ )
11 uzss 10498 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
123, 11syl 16 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
1312, 8, 13sstr4g 3381 . . . . . 6  |-  ( ph  ->  W  C_  Z )
1413sselda 3340 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  k  e.  Z )
1514, 6syldan 457 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  =  A )
1614, 7syldan 457 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  CC )
17 isumsplit.6 . . . . 5  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
186, 7eqeltrd 2509 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
191, 2, 18iserex 12442 . . . . 5  |-  ( ph  ->  (  seq  M (  +  ,  F )  e.  dom  ~~>  <->  seq  N (  +  ,  F )  e.  dom  ~~>  ) )
2017, 19mpbid 202 . . . 4  |-  ( ph  ->  seq  N (  +  ,  F )  e. 
dom 
~~>  )
218, 10, 15, 16, 20isumclim2 12534 . . 3  |-  ( ph  ->  seq  N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
22 fzfid 11304 . . . 4  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
23 elfzuz 11047 . . . . . 6  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
2423, 1syl6eleqr 2526 . . . . 5  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
2524, 7sylan2 461 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
2622, 25fsumcl 12519 . . 3  |-  ( ph  -> 
sum_ k  e.  ( M ... ( N  -  1 ) ) A  e.  CC )
2714, 18syldan 457 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  e.  CC )
288, 10, 27serf 11343 . . . 4  |-  ( ph  ->  seq  N (  +  ,  F ) : W --> CC )
2928ffvelrnda 5862 . . 3  |-  ( (
ph  /\  j  e.  W )  ->  (  seq  N (  +  ,  F ) `  j
)  e.  CC )
305zred 10367 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  RR )
3130ltm1d 9935 . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  1 )  <  M )
32 peano2zm 10312 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
335, 32syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
34 fzn 11063 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
355, 33, 34syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( ( M  - 
1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
3631, 35mpbid 202 . . . . . . . . . 10  |-  ( ph  ->  ( M ... ( M  -  1 ) )  =  (/) )
3736sumeq1d 12487 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  sum_ k  e.  (/)  A )
3837adantr 452 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  sum_ k  e.  (/)  A )
39 sum0 12507 . . . . . . . 8  |-  sum_ k  e.  (/)  A  =  0
4038, 39syl6eq 2483 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  0 )
4140oveq1d 6088 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  ( sum_ k  e.  ( M ... ( M  - 
1 ) ) A  +  (  seq  M
(  +  ,  F
) `  j )
)  =  ( 0  +  (  seq  M
(  +  ,  F
) `  j )
) )
4213sselda 3340 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  j  e.  Z )
431, 5, 18serf 11343 . . . . . . . . 9  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
4443ffvelrnda 5862 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  CC )
4542, 44syldan 457 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  CC )
4645addid2d 9259 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  (
0  +  (  seq 
M (  +  ,  F ) `  j
) )  =  (  seq  M (  +  ,  F ) `  j ) )
4741, 46eqtr2d 2468 . . . . 5  |-  ( (
ph  /\  j  e.  W )  ->  (  seq  M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( M  -  1 ) ) A  +  (  seq  M (  +  ,  F ) `  j ) ) )
48 oveq1 6080 . . . . . . . . 9  |-  ( N  =  M  ->  ( N  -  1 )  =  ( M  - 
1 ) )
4948oveq2d 6089 . . . . . . . 8  |-  ( N  =  M  ->  ( M ... ( N  - 
1 ) )  =  ( M ... ( M  -  1 ) ) )
5049sumeq1d 12487 . . . . . . 7  |-  ( N  =  M  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  sum_ k  e.  ( M ... ( M  -  1 ) ) A )
51 seqeq1 11318 . . . . . . . 8  |-  ( N  =  M  ->  seq  N (  +  ,  F
)  =  seq  M
(  +  ,  F
) )
5251fveq1d 5722 . . . . . . 7  |-  ( N  =  M  ->  (  seq  N (  +  ,  F ) `  j
)  =  (  seq 
M (  +  ,  F ) `  j
) )
5350, 52oveq12d 6091 . . . . . 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 ) ) )
5453eqeq2d 2446 . . . . 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 ) ) ) )
5547, 54syl5ibrcom 214 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  =  M  ->  (  seq  M (  +  ,  F ) `  j )  =  (
sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq 
N (  +  ,  F ) `  j
) ) ) )
56 addcl 9064 . . . . . . . 8  |-  ( ( k  e.  CC  /\  m  e.  CC )  ->  ( k  +  m
)  e.  CC )
5756adantl 453 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  ( k  e.  CC  /\  m  e.  CC ) )  -> 
( k  +  m
)  e.  CC )
58 addass 9069 . . . . . . . 8  |-  ( ( k  e.  CC  /\  m  e.  CC  /\  x  e.  CC )  ->  (
( k  +  m
)  +  x )  =  ( k  +  ( m  +  x
) ) )
5958adantl 453 . . . . . . 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 ) ) )
60 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  j  e.  W )
61 simpll 731 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ph )
6210zcnd 10368 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
63 ax-1cn 9040 . . . . . . . . . . . . 13  |-  1  e.  CC
64 npcan 9306 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
6562, 63, 64sylancl 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
6665eqcomd 2440 . . . . . . . . . . 11  |-  ( ph  ->  N  =  ( ( N  -  1 )  +  1 ) )
6761, 66syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  N  =  ( ( N  - 
1 )  +  1 ) )
6867fveq2d 5724 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( ZZ>= `  N )  =  (
ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
698, 68syl5eq 2479 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  W  =  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
7060, 69eleqtrd 2511 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  j  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
715adantr 452 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  M  e.  ZZ )
72 eluzp1m1 10501 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  M ) )
7371, 72sylan 458 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
74 elfzuz 11047 . . . . . . . . 9  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
7574, 1syl6eleqr 2526 . . . . . . . 8  |-  ( k  e.  ( M ... j )  ->  k  e.  Z )
7661, 75, 18syl2an 464 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... j ) )  ->  ( F `  k )  e.  CC )
7757, 59, 70, 73, 76seqsplit 11348 . . . . . 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
) ) )
7861, 24, 6syl2an 464 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  k )  =  A )
7961, 24, 7syl2an 464 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
8078, 73, 79fsumser 12516 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  (  seq  M (  +  ,  F ) `  ( N  -  1
) ) )
8167seqeq1d 11321 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  seq  N (  +  ,  F )  =  seq  ( ( N  -  1 )  +  1 ) (  +  ,  F ) )
8281fveq1d 5722 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  N (  +  ,  F
) `  j )  =  (  seq  ( ( N  -  1 )  +  1 ) (  +  ,  F ) `
 j ) )
8380, 82oveq12d 6091 . . . . . 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 )
) )
8477, 83eqtr4d 2470 . . . . 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 ) ) )
8584ex 424 . . . 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 ) ) ) )
86 uzp1 10511 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
873, 86syl 16 . . . . 5  |-  ( ph  ->  ( N  =  M  \/  N  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
8887adantr 452 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1
) ) ) )
8955, 85, 88mpjaod 371 . . 3  |-  ( (
ph  /\  j  e.  W )  ->  (  seq  M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq  N (  +  ,  F ) `  j ) ) )
908, 10, 21, 26, 17, 29, 89climaddc2 12421 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
911, 5, 6, 7, 90isumclim 12533 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312   (/)c0 3620   class class class wbr 4204   dom cdm 4870   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    < clt 9112    - cmin 9283   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315    ~~> cli 12270   sum_csu 12471
This theorem is referenced by:  isum1p  12613  geolim2  12640  mertenslem2  12654  mertens  12655  effsumlt  12704  eirrlem  12795  rpnnen2lem8  12813  prmreclem6  13281  aaliou3lem7  20258  abelthlem7  20346  log2tlbnd  20777  subfaclim  24866  stirlinglem12  27801
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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