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Theorem clim2ser 12403
Description: The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
Hypotheses
Ref Expression
clim2ser.1  |-  Z  =  ( ZZ>= `  M )
clim2ser.2  |-  ( ph  ->  N  e.  Z )
clim2ser.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
clim2ser.5  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  A )
Assertion
Ref Expression
clim2ser  |-  ( ph  ->  seq  ( N  + 
1 ) (  +  ,  F )  ~~>  ( A  -  (  seq  M
(  +  ,  F
) `  N )
) )
Distinct variable groups:    A, k    k, F    k, M    k, N    ph, k    k, Z

Proof of Theorem clim2ser
Dummy variables  j  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . 2  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2ser.2 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 clim2ser.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
42, 3syl6eleq 2494 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 peano2uz 10486 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
64, 5syl 16 . . 3  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
7 eluzelz 10452 . . 3  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ZZ )
86, 7syl 16 . 2  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
9 clim2ser.5 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  A )
10 eluzel2 10449 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
114, 10syl 16 . . . 4  |-  ( ph  ->  M  e.  ZZ )
12 clim2ser.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
133, 11, 12serf 11306 . . 3  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
1413, 2ffvelrnd 5830 . 2  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  e.  CC )
15 seqex 11280 . . 3  |-  seq  ( N  +  1 ) (  +  ,  F
)  e.  _V
1615a1i 11 . 2  |-  ( ph  ->  seq  ( N  + 
1 ) (  +  ,  F )  e. 
_V )
1713adantr 452 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  seq  M (  +  ,  F ) : Z --> CC )
186, 3syl6eleqr 2495 . . . 4  |-  ( ph  ->  ( N  +  1 )  e.  Z )
193uztrn2 10459 . . . 4  |-  ( ( ( N  +  1 )  e.  Z  /\  j  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
j  e.  Z )
2018, 19sylan 458 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  j  e.  Z )
2117, 20ffvelrnd 5830 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  j )  e.  CC )
22 addcl 9028 . . . . . 6  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  +  x
)  e.  CC )
2322adantl 453 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  +  x
)  e.  CC )
24 addass 9033 . . . . . 6  |-  ( ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
( k  +  x
)  +  y )  =  ( k  +  ( x  +  y ) ) )
2524adantl 453 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC ) )  -> 
( ( k  +  x )  +  y )  =  ( k  +  ( x  +  y ) ) )
26 simpr 448 . . . . 5  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  j  e.  ( ZZ>= `  ( N  +  1 ) ) )
274adantr 452 . . . . 5  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M )
)
28 elfzuz 11011 . . . . . . . 8  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
2928, 3syl6eleqr 2495 . . . . . . 7  |-  ( k  e.  ( M ... j )  ->  k  e.  Z )
3029, 12sylan2 461 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... j ) )  ->  ( F `  k )  e.  CC )
3130adantlr 696 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( M ... j
) )  ->  ( F `  k )  e.  CC )
3223, 25, 26, 27, 31seqsplit 11311 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  j )  =  ( (  seq 
M (  +  ,  F ) `  N
)  +  (  seq  ( N  +  1 ) (  +  ,  F ) `  j
) ) )
3332oveq1d 6055 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( (  seq  M (  +  ,  F ) `  j
)  -  (  seq 
M (  +  ,  F ) `  N
) )  =  ( ( (  seq  M
(  +  ,  F
) `  N )  +  (  seq  ( N  +  1 ) (  +  ,  F ) `
 j ) )  -  (  seq  M
(  +  ,  F
) `  N )
) )
3414adantr 452 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  N )  e.  CC )
353uztrn2 10459 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
3618, 35sylan 458 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  Z )
3736, 12syldan 457 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  e.  CC )
381, 8, 37serf 11306 . . . . 5  |-  ( ph  ->  seq  ( N  + 
1 ) (  +  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
3938ffvelrnda 5829 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  +  ,  F ) `  j
)  e.  CC )
4034, 39pncan2d 9369 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( (
(  seq  M (  +  ,  F ) `  N )  +  (  seq  ( N  + 
1 ) (  +  ,  F ) `  j ) )  -  (  seq  M (  +  ,  F ) `  N ) )  =  (  seq  ( N  +  1 ) (  +  ,  F ) `
 j ) )
4133, 40eqtr2d 2437 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  +  ,  F ) `  j
)  =  ( (  seq  M (  +  ,  F ) `  j )  -  (  seq  M (  +  ,  F ) `  N
) ) )
421, 8, 9, 14, 16, 21, 41climsubc1 12386 1  |-  ( ph  ->  seq  ( N  + 
1 ) (  +  ,  F )  ~~>  ( A  -  (  seq  M
(  +  ,  F
) `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   class class class wbr 4172   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   1c1 8947    + caddc 8949    - cmin 9247   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278    ~~> cli 12233
This theorem is referenced by:  iserex  12405  ege2le3  12647  abelthlem9  20309  stirlinglem7  27696  stirlinglem11  27700  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-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-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-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  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-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237
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