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Theorem isumsplit 12575
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 2494 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 eluzel2 10449 . . 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 10452 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
103, 9syl 16 . . 3  |-  ( ph  ->  N  e.  ZZ )
11 uzss 10462 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
123, 11syl 16 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
1312, 8, 13sstr4g 3349 . . . . . 6  |-  ( ph  ->  W  C_  Z )
1413sselda 3308 . . . . 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 2478 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
191, 2, 18iserex 12405 . . . . 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 12497 . . 3  |-  ( ph  ->  seq  N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
22 fzfid 11267 . . . 4  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
23 elfzuz 11011 . . . . . 6  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
2423, 1syl6eleqr 2495 . . . . 5  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
2524, 7sylan2 461 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
2622, 25fsumcl 12482 . . 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 11306 . . . 4  |-  ( ph  ->  seq  N (  +  ,  F ) : W --> CC )
2928ffvelrnda 5829 . . 3  |-  ( (
ph  /\  j  e.  W )  ->  (  seq  N (  +  ,  F ) `  j
)  e.  CC )
305zred 10331 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  RR )
3130ltm1d 9899 . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  1 )  <  M )
32 peano2zm 10276 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
335, 32syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
34 fzn 11027 . . . . . . . . . . . 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 12450 . . . . . . . . 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 12470 . . . . . . . 8  |-  sum_ k  e.  (/)  A  =  0
4038, 39syl6eq 2452 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  0 )
4140oveq1d 6055 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  ( sum_ k  e.  ( M ... ( M  - 
1 ) ) A  +  (  seq  M
(  +  ,  F
) `  j )
)  =  ( 0  +  (  seq  M
(  +  ,  F
) `  j )
) )
4213sselda 3308 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  j  e.  Z )
431, 5, 18serf 11306 . . . . . . . . 9  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
4443ffvelrnda 5829 . . . . . . . 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 9223 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  (
0  +  (  seq 
M (  +  ,  F ) `  j
) )  =  (  seq  M (  +  ,  F ) `  j ) )
4741, 46eqtr2d 2437 . . . . 5  |-  ( (
ph  /\  j  e.  W )  ->  (  seq  M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( M  -  1 ) ) A  +  (  seq  M (  +  ,  F ) `  j ) ) )
48 oveq1 6047 . . . . . . . . 9  |-  ( N  =  M  ->  ( N  -  1 )  =  ( M  - 
1 ) )
4948oveq2d 6056 . . . . . . . 8  |-  ( N  =  M  ->  ( M ... ( N  - 
1 ) )  =  ( M ... ( M  -  1 ) ) )
5049sumeq1d 12450 . . . . . . 7  |-  ( N  =  M  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  sum_ k  e.  ( M ... ( M  -  1 ) ) A )
51 seqeq1 11281 . . . . . . . 8  |-  ( N  =  M  ->  seq  N (  +  ,  F
)  =  seq  M
(  +  ,  F
) )
5251fveq1d 5689 . . . . . . 7  |-  ( N  =  M  ->  (  seq  N (  +  ,  F ) `  j
)  =  (  seq 
M (  +  ,  F ) `  j
) )
5350, 52oveq12d 6058 . . . . . 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 2415 . . . . 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 9028 . . . . . . . 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 9033 . . . . . . . 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 10332 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
63 ax-1cn 9004 . . . . . . . . . . . . 13  |-  1  e.  CC
64 npcan 9270 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
6562, 63, 64sylancl 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
6665eqcomd 2409 . . . . . . . . . . 11  |-  ( ph  ->  N  =  ( ( N  -  1 )  +  1 ) )
6761, 66syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  N  =  ( ( N  - 
1 )  +  1 ) )
6867fveq2d 5691 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( ZZ>= `  N )  =  (
ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
698, 68syl5eq 2448 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  W  =  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
7060, 69eleqtrd 2480 . . . . . . 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 10465 . . . . . . . 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 11011 . . . . . . . . 9  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
7574, 1syl6eleqr 2495 . . . . . . . 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 11311 . . . . . 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 12479 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  (  seq  M (  +  ,  F ) `  ( N  -  1
) ) )
8167seqeq1d 11284 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  seq  N (  +  ,  F )  =  seq  ( ( N  -  1 )  +  1 ) (  +  ,  F ) )
8281fveq1d 5689 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  N (  +  ,  F
) `  j )  =  (  seq  ( ( N  -  1 )  +  1 ) (  +  ,  F ) `
 j ) )
8380, 82oveq12d 6058 . . . . . 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 2439 . . . . 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 10475 . . . . . 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 12384 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
911, 5, 6, 7, 90isumclim 12496 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 1649    e. wcel 1721    C_ wss 3280   (/)c0 3588   class class class wbr 4172   dom cdm 4837   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    < clt 9076    - cmin 9247   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278    ~~> cli 12233   sum_csu 12434
This theorem is referenced by:  isum1p  12576  geolim2  12603  mertenslem2  12617  mertens  12618  effsumlt  12667  eirrlem  12758  rpnnen2lem8  12776  prmreclem6  13244  aaliou3lem7  20219  abelthlem7  20307  log2tlbnd  20738  subfaclim  24827  stirlinglem12  27701
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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