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Theorem isum1p 13300
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 2441 . . 3  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  ( M  + 
1 ) )
3 isum1p.3 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
4 uzid 10871 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
53, 4syl 16 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
6 peano2uz 10904 . . . . 5  |-  ( M  e.  ( ZZ>= `  M
)  ->  ( M  +  1 )  e.  ( ZZ>= `  M )
)
75, 6syl 16 . . . 4  |-  ( ph  ->  ( M  +  1 )  e.  ( ZZ>= `  M ) )
87, 1syl6eleqr 2532 . . 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 13299 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( sum_ k  e.  ( M ... ( ( M  + 
1 )  -  1 ) ) A  +  sum_ k  e.  ( ZZ>= `  ( M  +  1
) ) A ) )
133zcnd 10744 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
14 ax-1cn 9336 . . . . . . 7  |-  1  e.  CC
15 pncan 9612 . . . . . . 7  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  + 
1 )  -  1 )  =  M )
1613, 14, 15sylancl 657 . . . . . 6  |-  ( ph  ->  ( ( M  + 
1 )  -  1 )  =  M )
1716oveq2d 6106 . . . . 5  |-  ( ph  ->  ( M ... (
( M  +  1 )  -  1 ) )  =  ( M ... M ) )
1817sumeq1d 13174 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... ( ( M  +  1 )  -  1 ) ) A  =  sum_ k  e.  ( M ... M
) A )
19 elfzuz 11445 . . . . . . 7  |-  ( k  e.  ( M ... M )  ->  k  e.  ( ZZ>= `  M )
)
2019, 1syl6eleqr 2532 . . . . . 6  |-  ( k  e.  ( M ... M )  ->  k  e.  Z )
2120, 9sylan2 471 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... M ) )  ->  ( F `  k )  =  A )
2221sumeq2dv 13176 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... M ) ( F `  k
)  =  sum_ k  e.  ( M ... M
) A )
235, 1syl6eleqr 2532 . . . . . 6  |-  ( ph  ->  M  e.  Z )
249, 10eqeltrd 2515 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2524ralrimiva 2797 . . . . . 6  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
26 fveq2 5688 . . . . . . . 8  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2726eleq1d 2507 . . . . . . 7  |-  ( k  =  M  ->  (
( F `  k
)  e.  CC  <->  ( F `  M )  e.  CC ) )
2827rspcv 3066 . . . . . 6  |-  ( M  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  CC  ->  ( F `  M )  e.  CC ) )
2923, 25, 28sylc 60 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  CC )
3026fsum1 13214 . . . . 5  |-  ( ( M  e.  ZZ  /\  ( F `  M )  e.  CC )  ->  sum_ k  e.  ( M ... M ) ( F `  k )  =  ( F `  M ) )
313, 29, 30syl2anc 656 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... M ) ( F `  k
)  =  ( F `
 M ) )
3218, 22, 313eqtr2d 2479 . . 3  |-  ( ph  -> 
sum_ k  e.  ( M ... ( ( M  +  1 )  -  1 ) ) A  =  ( F `
 M ) )
3332oveq1d 6105 . 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 2473 1  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( ( F `  M )  +  sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   dom cdm 4836   ` cfv 5415  (class class class)co 6090   CCcc 9276   1c1 9279    + caddc 9281    - cmin 9591   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433    seqcseq 11802    ~~> cli 12958   sum_csu 13159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160
This theorem is referenced by:  isumnn0nn  13301  efsep  13390  rpnnen2lem9  13501
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