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Theorem isumsplit 13286
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 2523 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 eluzel2 10854 . . 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 10858 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
103, 9syl 16 . . 3  |-  ( ph  ->  N  e.  ZZ )
11 uzss 10869 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
123, 11syl 16 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
1312, 8, 13sstr4g 3385 . . . . . 6  |-  ( ph  ->  W  C_  Z )
1413sselda 3344 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  k  e.  Z )
1514, 6syldan 467 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  =  A )
1614, 7syldan 467 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  CC )
17 isumsplit.6 . . . . 5  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
186, 7eqeltrd 2507 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
191, 2, 18iserex 13118 . . . . 5  |-  ( ph  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
2017, 19mpbid 210 . . . 4  |-  ( ph  ->  seq N (  +  ,  F )  e. 
dom 
~~>  )
218, 10, 15, 16, 20isumclim2 13209 . . 3  |-  ( ph  ->  seq N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
22 fzfid 11779 . . . 4  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
23 elfzuz 11436 . . . . . 6  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
2423, 1syl6eleqr 2524 . . . . 5  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
2524, 7sylan2 471 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
2622, 25fsumcl 13194 . . 3  |-  ( ph  -> 
sum_ k  e.  ( M ... ( N  -  1 ) ) A  e.  CC )
2714, 18syldan 467 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  e.  CC )
288, 10, 27serf 11818 . . . 4  |-  ( ph  ->  seq N (  +  ,  F ) : W --> CC )
2928ffvelrnda 5831 . . 3  |-  ( (
ph  /\  j  e.  W )  ->  (  seq N (  +  ,  F ) `  j
)  e.  CC )
305zred 10735 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  RR )
3130ltm1d 10253 . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  1 )  <  M )
32 peano2zm 10676 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
335, 32syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
34 fzn 11453 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
355, 33, 34syl2anc 654 . . . . . . . . . . 11  |-  ( ph  ->  ( ( M  - 
1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
3631, 35mpbid 210 . . . . . . . . . 10  |-  ( ph  ->  ( M ... ( M  -  1 ) )  =  (/) )
3736sumeq1d 13162 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  sum_ k  e.  (/)  A )
3837adantr 462 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  sum_ k  e.  (/)  A )
39 sum0 13182 . . . . . . . 8  |-  sum_ k  e.  (/)  A  =  0
4038, 39syl6eq 2481 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  0 )
4140oveq1d 6095 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  ( sum_ k  e.  ( M ... ( M  - 
1 ) ) A  +  (  seq M
(  +  ,  F
) `  j )
)  =  ( 0  +  (  seq M
(  +  ,  F
) `  j )
) )
4213sselda 3344 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  j  e.  Z )
431, 5, 18serf 11818 . . . . . . . . 9  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
4443ffvelrnda 5831 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  e.  CC )
4542, 44syldan 467 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  (  seq M (  +  ,  F ) `  j
)  e.  CC )
4645addid2d 9558 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  (
0  +  (  seq M (  +  ,  F ) `  j
) )  =  (  seq M (  +  ,  F ) `  j ) )
4741, 46eqtr2d 2466 . . . . 5  |-  ( (
ph  /\  j  e.  W )  ->  (  seq M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( M  -  1 ) ) A  +  (  seq M (  +  ,  F ) `  j ) ) )
48 oveq1 6087 . . . . . . . . 9  |-  ( N  =  M  ->  ( N  -  1 )  =  ( M  - 
1 ) )
4948oveq2d 6096 . . . . . . . 8  |-  ( N  =  M  ->  ( M ... ( N  - 
1 ) )  =  ( M ... ( M  -  1 ) ) )
5049sumeq1d 13162 . . . . . . 7  |-  ( N  =  M  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  sum_ k  e.  ( M ... ( M  -  1 ) ) A )
51 seqeq1 11793 . . . . . . . 8  |-  ( N  =  M  ->  seq N (  +  ,  F )  =  seq M (  +  ,  F ) )
5251fveq1d 5681 . . . . . . 7  |-  ( N  =  M  ->  (  seq N (  +  ,  F ) `  j
)  =  (  seq M (  +  ,  F ) `  j
) )
5350, 52oveq12d 6098 . . . . . 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 2444 . . . . 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 222 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  =  M  ->  (  seq M (  +  ,  F ) `  j )  =  (
sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq N (  +  ,  F ) `  j
) ) ) )
56 addcl 9352 . . . . . . . 8  |-  ( ( k  e.  CC  /\  m  e.  CC )  ->  ( k  +  m
)  e.  CC )
5756adantl 463 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  ( k  e.  CC  /\  m  e.  CC ) )  -> 
( k  +  m
)  e.  CC )
58 addass 9357 . . . . . . . 8  |-  ( ( k  e.  CC  /\  m  e.  CC  /\  x  e.  CC )  ->  (
( k  +  m
)  +  x )  =  ( k  +  ( m  +  x
) ) )
5958adantl 463 . . . . . . 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 747 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  j  e.  W )
61 simpll 746 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ph )
6210zcnd 10736 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
63 ax-1cn 9328 . . . . . . . . . . . . 13  |-  1  e.  CC
64 npcan 9607 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
6562, 63, 64sylancl 655 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
6665eqcomd 2438 . . . . . . . . . . 11  |-  ( ph  ->  N  =  ( ( N  -  1 )  +  1 ) )
6761, 66syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  N  =  ( ( N  - 
1 )  +  1 ) )
6867fveq2d 5683 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( ZZ>= `  N )  =  (
ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
698, 68syl5eq 2477 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  W  =  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
7060, 69eleqtrd 2509 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  j  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
715adantr 462 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  M  e.  ZZ )
72 eluzp1m1 10872 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  M ) )
7371, 72sylan 468 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
74 elfzuz 11436 . . . . . . . . 9  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
7574, 1syl6eleqr 2524 . . . . . . . 8  |-  ( k  e.  ( M ... j )  ->  k  e.  Z )
7661, 75, 18syl2an 474 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... j ) )  ->  ( F `  k )  e.  CC )
7757, 59, 70, 73, 76seqsplit 11823 . . . . . 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 474 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  k )  =  A )
7961, 24, 7syl2an 474 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
8078, 73, 79fsumser 13191 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  (  seq M (  +  ,  F ) `  ( N  -  1
) ) )
8167seqeq1d 11796 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  seq N (  +  ,  F )  =  seq ( ( N  -  1 )  +  1 ) (  +  ,  F ) )
8281fveq1d 5681 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq N (  +  ,  F ) `  j
)  =  (  seq ( ( N  - 
1 )  +  1 ) (  +  ,  F ) `  j
) )
8380, 82oveq12d 6098 . . . . . 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 2468 . . . . 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 434 . . . 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 10882 . . . . . 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 462 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1
) ) ) )
8955, 85, 88mpjaod 381 . . 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 13097 . 2  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
911, 5, 6, 7, 90isumclim 13208 1  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755    C_ wss 3316   (/)c0 3625   class class class wbr 4280   dom cdm 4827   ` cfv 5406  (class class class)co 6080   CCcc 9268   0cc0 9270   1c1 9271    + caddc 9273    < clt 9406    - cmin 9583   ZZcz 10634   ZZ>=cuz 10849   ...cfz 11424    seqcseq 11790    ~~> cli 12946   sum_csu 13147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148
This theorem is referenced by:  isum1p  13287  geolim2  13314  mertenslem2  13328  mertens  13329  effsumlt  13378  eirrlem  13469  rpnnen2lem8  13487  prmreclem6  13965  aaliou3lem7  21700  abelthlem7  21788  log2tlbnd  22225  subfaclim  26924  stirlinglem12  29726
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