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Theorem isum1p 12300
Description: The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
isum1p.1  |-  Z  =  ( ZZ>= `  M )
isum1p.3  |-  ( ph  ->  M  e.  ZZ )
isum1p.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
isum1p.5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
isum1p.6  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
Assertion
Ref Expression
isum1p  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( ( F `  M )  +  sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) A ) )
Distinct variable groups:    k, F    k, M    ph, k    k, Z
Allowed substitution hint:    A( k)

Proof of Theorem isum1p
StepHypRef Expression
1 isum1p.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 eqid 2283 . . 3  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  ( M  + 
1 ) )
3 isum1p.3 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
4 uzid 10242 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
53, 4syl 15 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
6 peano2uz 10272 . . . . 5  |-  ( M  e.  ( ZZ>= `  M
)  ->  ( M  +  1 )  e.  ( ZZ>= `  M )
)
75, 6syl 15 . . . 4  |-  ( ph  ->  ( M  +  1 )  e.  ( ZZ>= `  M ) )
87, 1syl6eleqr 2374 . . 3  |-  ( ph  ->  ( M  +  1 )  e.  Z )
9 isum1p.4 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
10 isum1p.5 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
11 isum1p.6 . . 3  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
121, 2, 8, 9, 10, 11isumsplit 12299 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( sum_ k  e.  ( M ... ( ( M  + 
1 )  -  1 ) ) A  +  sum_ k  e.  ( ZZ>= `  ( M  +  1
) ) A ) )
133zcnd 10118 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
14 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
15 pncan 9057 . . . . . . 7  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  + 
1 )  -  1 )  =  M )
1613, 14, 15sylancl 643 . . . . . 6  |-  ( ph  ->  ( ( M  + 
1 )  -  1 )  =  M )
1716oveq2d 5874 . . . . 5  |-  ( ph  ->  ( M ... (
( M  +  1 )  -  1 ) )  =  ( M ... M ) )
1817sumeq1d 12174 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... ( ( M  +  1 )  -  1 ) ) A  =  sum_ k  e.  ( M ... M
) A )
19 elfzuz 10794 . . . . . . 7  |-  ( k  e.  ( M ... M )  ->  k  e.  ( ZZ>= `  M )
)
2019, 1syl6eleqr 2374 . . . . . 6  |-  ( k  e.  ( M ... M )  ->  k  e.  Z )
2120, 9sylan2 460 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... M ) )  ->  ( F `  k )  =  A )
2221sumeq2dv 12176 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... M ) ( F `  k
)  =  sum_ k  e.  ( M ... M
) A )
235, 1syl6eleqr 2374 . . . . . 6  |-  ( ph  ->  M  e.  Z )
249, 10eqeltrd 2357 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2524ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
26 fveq2 5525 . . . . . . . 8  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2726eleq1d 2349 . . . . . . 7  |-  ( k  =  M  ->  (
( F `  k
)  e.  CC  <->  ( F `  M )  e.  CC ) )
2827rspcv 2880 . . . . . 6  |-  ( M  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  CC  ->  ( F `  M )  e.  CC ) )
2923, 25, 28sylc 56 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  CC )
3026fsum1 12214 . . . . 5  |-  ( ( M  e.  ZZ  /\  ( F `  M )  e.  CC )  ->  sum_ k  e.  ( M ... M ) ( F `  k )  =  ( F `  M ) )
313, 29, 30syl2anc 642 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... M ) ( F `  k
)  =  ( F `
 M ) )
3218, 22, 313eqtr2d 2321 . . 3  |-  ( ph  -> 
sum_ k  e.  ( M ... ( ( M  +  1 )  -  1 ) ) A  =  ( F `
 M ) )
3332oveq1d 5873 . 2  |-  ( ph  ->  ( sum_ k  e.  ( M ... ( ( M  +  1 )  -  1 ) ) A  +  sum_ k  e.  ( ZZ>= `  ( M  +  1 ) ) A )  =  ( ( F `  M
)  +  sum_ k  e.  ( ZZ>= `  ( M  +  1 ) ) A ) )
3412, 33eqtrd 2315 1  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( ( F `  M )  +  sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   dom cdm 4689   ` 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   sum_csu 12158
This theorem is referenced by:  isumnn0nn  12301  efsep  12390  rpnnen2lem9  12501
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-int 3863  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-se 4353  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-isom 5264  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-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  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-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159
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