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Theorem clim2ser 12377
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 2389 . 2  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2ser.2 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 clim2ser.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
42, 3syl6eleq 2479 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 peano2uz 10464 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
64, 5syl 16 . . 3  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
7 eluzelz 10430 . . 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 10427 . . . . 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 11280 . . 3  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
1413, 2ffvelrnd 5812 . 2  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  e.  CC )
15 seqex 11254 . . 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 2480 . . . 4  |-  ( ph  ->  ( N  +  1 )  e.  Z )
193uztrn2 10437 . . . 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 5812 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  j )  e.  CC )
22 addcl 9007 . . . . . 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 9012 . . . . . 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 10989 . . . . . . . 8  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
2928, 3syl6eleqr 2480 . . . . . . 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 11285 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  j )  =  ( (  seq 
M (  +  ,  F ) `  N
)  +  (  seq  ( N  +  1 ) (  +  ,  F ) `  j
) ) )
3332oveq1d 6037 . . 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 10437 . . . . . . . 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 11280 . . . . 5  |-  ( ph  ->  seq  ( N  + 
1 ) (  +  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
3938ffvelrnda 5811 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  +  ,  F ) `  j
)  e.  CC )
4034, 39pncan2d 9347 . . 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 2422 . 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 12360 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 1717   _Vcvv 2901   class class class wbr 4155   -->wf 5392   ` cfv 5396  (class class class)co 6022   CCcc 8923   1c1 8926    + caddc 8928    - cmin 9225   ZZcz 10216   ZZ>=cuz 10422   ...cfz 10977    seq cseq 11252    ~~> cli 12207
This theorem is referenced by:  iserex  12379  ege2le3  12621  abelthlem9  20225  stirlinglem7  27499  stirlinglem11  27503  stirlinglem12  27504
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-fz 10978  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211
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