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Theorem isumsplit 13654
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 2480 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 eluzel2 11006 . . 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 11010 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
103, 9syl 16 . . 3  |-  ( ph  ->  N  e.  ZZ )
11 uzss 11021 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
123, 11syl 16 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
1312, 8, 13sstr4g 3458 . . . . . 6  |-  ( ph  ->  W  C_  Z )
1413sselda 3417 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  k  e.  Z )
1514, 6syldan 468 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  =  A )
1614, 7syldan 468 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  CC )
17 isumsplit.6 . . . . 5  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
186, 7eqeltrd 2470 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
191, 2, 18iserex 13481 . . . . 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 13575 . . 3  |-  ( ph  ->  seq N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
22 fzfid 11986 . . . 4  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
23 elfzuz 11605 . . . . . 6  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
2423, 1syl6eleqr 2481 . . . . 5  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
2524, 7sylan2 472 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
2622, 25fsumcl 13557 . . 3  |-  ( ph  -> 
sum_ k  e.  ( M ... ( N  -  1 ) ) A  e.  CC )
2714, 18syldan 468 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  e.  CC )
288, 10, 27serf 12038 . . . 4  |-  ( ph  ->  seq N (  +  ,  F ) : W --> CC )
2928ffvelrnda 5933 . . 3  |-  ( (
ph  /\  j  e.  W )  ->  (  seq N (  +  ,  F ) `  j
)  e.  CC )
305zred 10884 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  RR )
3130ltm1d 10394 . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  1 )  <  M )
32 peano2zm 10824 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
335, 32syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
34 fzn 11623 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
355, 33, 34syl2anc 659 . . . . . . . . . . 11  |-  ( ph  ->  ( ( M  - 
1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
3631, 35mpbid 210 . . . . . . . . . 10  |-  ( ph  ->  ( M ... ( M  -  1 ) )  =  (/) )
3736sumeq1d 13525 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  sum_ k  e.  (/)  A )
3837adantr 463 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  sum_ k  e.  (/)  A )
39 sum0 13545 . . . . . . . 8  |-  sum_ k  e.  (/)  A  =  0
4038, 39syl6eq 2439 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  0 )
4140oveq1d 6211 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  ( sum_ k  e.  ( M ... ( M  - 
1 ) ) A  +  (  seq M
(  +  ,  F
) `  j )
)  =  ( 0  +  (  seq M
(  +  ,  F
) `  j )
) )
4213sselda 3417 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  j  e.  Z )
431, 5, 18serf 12038 . . . . . . . . 9  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
4443ffvelrnda 5933 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  e.  CC )
4542, 44syldan 468 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  (  seq M (  +  ,  F ) `  j
)  e.  CC )
4645addid2d 9692 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  (
0  +  (  seq M (  +  ,  F ) `  j
) )  =  (  seq M (  +  ,  F ) `  j ) )
4741, 46eqtr2d 2424 . . . . 5  |-  ( (
ph  /\  j  e.  W )  ->  (  seq M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( M  -  1 ) ) A  +  (  seq M (  +  ,  F ) `  j ) ) )
48 oveq1 6203 . . . . . . . . 9  |-  ( N  =  M  ->  ( N  -  1 )  =  ( M  - 
1 ) )
4948oveq2d 6212 . . . . . . . 8  |-  ( N  =  M  ->  ( M ... ( N  - 
1 ) )  =  ( M ... ( M  -  1 ) ) )
5049sumeq1d 13525 . . . . . . 7  |-  ( N  =  M  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  sum_ k  e.  ( M ... ( M  -  1 ) ) A )
51 seqeq1 12013 . . . . . . . 8  |-  ( N  =  M  ->  seq N (  +  ,  F )  =  seq M (  +  ,  F ) )
5251fveq1d 5776 . . . . . . 7  |-  ( N  =  M  ->  (  seq N (  +  ,  F ) `  j
)  =  (  seq M (  +  ,  F ) `  j
) )
5350, 52oveq12d 6214 . . . . . 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 2396 . . . . 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 9485 . . . . . . . 8  |-  ( ( k  e.  CC  /\  m  e.  CC )  ->  ( k  +  m
)  e.  CC )
5756adantl 464 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  ( k  e.  CC  /\  m  e.  CC ) )  -> 
( k  +  m
)  e.  CC )
58 addass 9490 . . . . . . . 8  |-  ( ( k  e.  CC  /\  m  e.  CC  /\  x  e.  CC )  ->  (
( k  +  m
)  +  x )  =  ( k  +  ( m  +  x
) ) )
5958adantl 464 . . . . . . 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 753 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  j  e.  W )
61 simpll 751 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ph )
6210zcnd 10885 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
63 ax-1cn 9461 . . . . . . . . . . . . 13  |-  1  e.  CC
64 npcan 9742 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
6562, 63, 64sylancl 660 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
6665eqcomd 2390 . . . . . . . . . . 11  |-  ( ph  ->  N  =  ( ( N  -  1 )  +  1 ) )
6761, 66syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  N  =  ( ( N  - 
1 )  +  1 ) )
6867fveq2d 5778 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( ZZ>= `  N )  =  (
ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
698, 68syl5eq 2435 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  W  =  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
7060, 69eleqtrd 2472 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  j  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
715adantr 463 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  M  e.  ZZ )
72 eluzp1m1 11024 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  M ) )
7371, 72sylan 469 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
74 elfzuz 11605 . . . . . . . . 9  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
7574, 1syl6eleqr 2481 . . . . . . . 8  |-  ( k  e.  ( M ... j )  ->  k  e.  Z )
7661, 75, 18syl2an 475 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... j ) )  ->  ( F `  k )  e.  CC )
7757, 59, 70, 73, 76seqsplit 12043 . . . . . 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 475 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  k )  =  A )
7961, 24, 7syl2an 475 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
8078, 73, 79fsumser 13554 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  (  seq M (  +  ,  F ) `  ( N  -  1
) ) )
8167seqeq1d 12016 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  seq N (  +  ,  F )  =  seq ( ( N  -  1 )  +  1 ) (  +  ,  F ) )
8281fveq1d 5776 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq N (  +  ,  F ) `  j
)  =  (  seq ( ( N  - 
1 )  +  1 ) (  +  ,  F ) `  j
) )
8380, 82oveq12d 6214 . . . . . 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 2426 . . . . 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 432 . . . 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 11034 . . . . . 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 463 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1
) ) ) )
8955, 85, 88mpjaod 379 . . 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 13460 . 2  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
911, 5, 6, 7, 90isumclim 13574 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 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    C_ wss 3389   (/)c0 3711   class class class wbr 4367   dom cdm 4913   ` cfv 5496  (class class class)co 6196   CCcc 9401   0cc0 9403   1c1 9404    + caddc 9406    < clt 9539    - cmin 9718   ZZcz 10781   ZZ>=cuz 11001   ...cfz 11593    seqcseq 12010    ~~> cli 13309   sum_csu 13510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-sum 13511
This theorem is referenced by:  isum1p  13655  geolim2  13682  mertenslem2  13696  mertens  13697  effsumlt  13848  eirrlem  13939  rpnnen2lem8  13957  prmreclem6  14441  aaliou3lem7  22830  abelthlem7  22918  log2tlbnd  23392  subfaclim  28821  binomcxplemnn0  31422  stirlinglem12  32033
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