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Theorem clim2ser 12128
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 2283 . 2  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2ser.2 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 clim2ser.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
42, 3syl6eleq 2373 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 peano2uz 10272 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
64, 5syl 15 . . 3  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
7 eluzelz 10238 . . 3  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ZZ )
86, 7syl 15 . 2  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
9 clim2ser.5 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  A )
10 eluzel2 10235 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
114, 10syl 15 . . . 4  |-  ( ph  ->  M  e.  ZZ )
12 clim2ser.4 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
133, 11, 12serf 11074 . . 3  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
14 ffvelrn 5663 . . 3  |-  ( (  seq  M (  +  ,  F ) : Z --> CC  /\  N  e.  Z )  ->  (  seq  M (  +  ,  F ) `  N
)  e.  CC )
1513, 2, 14syl2anc 642 . 2  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  e.  CC )
16 seqex 11048 . . 3  |-  seq  ( N  +  1 ) (  +  ,  F
)  e.  _V
1716a1i 10 . 2  |-  ( ph  ->  seq  ( N  + 
1 ) (  +  ,  F )  e. 
_V )
1813adantr 451 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  seq  M (  +  ,  F ) : Z --> CC )
196, 3syl6eleqr 2374 . . . 4  |-  ( ph  ->  ( N  +  1 )  e.  Z )
203uztrn2 10245 . . . 4  |-  ( ( ( N  +  1 )  e.  Z  /\  j  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
j  e.  Z )
2119, 20sylan 457 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  j  e.  Z )
22 ffvelrn 5663 . . 3  |-  ( (  seq  M (  +  ,  F ) : Z --> CC  /\  j  e.  Z )  ->  (  seq  M (  +  ,  F ) `  j
)  e.  CC )
2318, 21, 22syl2anc 642 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  j )  e.  CC )
24 addcl 8819 . . . . . 6  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  +  x
)  e.  CC )
2524adantl 452 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  +  x
)  e.  CC )
26 addass 8824 . . . . . 6  |-  ( ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
( k  +  x
)  +  y )  =  ( k  +  ( x  +  y ) ) )
2726adantl 452 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC ) )  -> 
( ( k  +  x )  +  y )  =  ( k  +  ( x  +  y ) ) )
28 simpr 447 . . . . 5  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  j  e.  ( ZZ>= `  ( N  +  1 ) ) )
294adantr 451 . . . . 5  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M )
)
30 elfzuz 10794 . . . . . . . 8  |-  ( k  e.  ( M ... j )  ->  k  e.  ( ZZ>= `  M )
)
3130, 3syl6eleqr 2374 . . . . . . 7  |-  ( k  e.  ( M ... j )  ->  k  e.  Z )
3231, 12sylan2 460 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... j ) )  ->  ( F `  k )  e.  CC )
3332adantlr 695 . . . . 5  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( M ... j
) )  ->  ( F `  k )  e.  CC )
3425, 27, 28, 29, 33seqsplit 11079 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  j )  =  ( (  seq 
M (  +  ,  F ) `  N
)  +  (  seq  ( N  +  1 ) (  +  ,  F ) `  j
) ) )
3534oveq1d 5873 . . 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 )
) )
3615adantr 451 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  +  ,  F
) `  N )  e.  CC )
373uztrn2 10245 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
3819, 37sylan 457 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  Z )
3938, 12syldan 456 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  e.  CC )
401, 8, 39serf 11074 . . . . 5  |-  ( ph  ->  seq  ( N  + 
1 ) (  +  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
41 ffvelrn 5663 . . . . 5  |-  ( (  seq  ( N  + 
1 ) (  +  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  +  ,  F ) `  j
)  e.  CC )
4240, 41sylan 457 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  +  ,  F ) `  j
)  e.  CC )
4336, 42pncan2d 9159 . . 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 ) )
4435, 43eqtr2d 2316 . 2  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  +  ,  F ) `  j
)  =  ( (  seq  M (  +  ,  F ) `  j )  -  (  seq  M (  +  ,  F ) `  N
) ) )
451, 8, 9, 15, 17, 23, 44climsubc1 12111 1  |-  ( ph  ->  seq  ( N  + 
1 ) (  +  ,  F )  ~~>  ( A  -  (  seq  M
(  +  ,  F
) `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046    ~~> cli 11958
This theorem is referenced by:  iserex  12130  ege2le3  12371  abelthlem9  19816  stirlinglem7  27829  stirlinglem11  27833  stirlinglem12  27834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962
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