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Theorem isum1p 11201
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 2115 . . 3  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  ( M  + 
1 ) )
3 isum1p.3 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
4 uzid 9289 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
53, 4syl 14 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
6 peano2uz 9327 . . . . 5  |-  ( M  e.  ( ZZ>= `  M
)  ->  ( M  +  1 )  e.  ( ZZ>= `  M )
)
75, 6syl 14 . . . 4  |-  ( ph  ->  ( M  +  1 )  e.  ( ZZ>= `  M ) )
87, 1syl6eleqr 2209 . . 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 11200 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( sum_ k  e.  ( M ... ( ( M  + 
1 )  -  1 ) ) A  +  sum_ k  e.  ( ZZ>= `  ( M  +  1
) ) A ) )
133zcnd 9125 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
14 ax-1cn 7677 . . . . . . 7  |-  1  e.  CC
15 pncan 7932 . . . . . . 7  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  + 
1 )  -  1 )  =  M )
1613, 14, 15sylancl 407 . . . . . 6  |-  ( ph  ->  ( ( M  + 
1 )  -  1 )  =  M )
1716oveq2d 5756 . . . . 5  |-  ( ph  ->  ( M ... (
( M  +  1 )  -  1 ) )  =  ( M ... M ) )
1817sumeq1d 11075 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... ( ( M  +  1 )  -  1 ) ) A  =  sum_ k  e.  ( M ... M
) A )
19 elfzuz 9742 . . . . . . 7  |-  ( k  e.  ( M ... M )  ->  k  e.  ( ZZ>= `  M )
)
2019, 1syl6eleqr 2209 . . . . . 6  |-  ( k  e.  ( M ... M )  ->  k  e.  Z )
2120, 9sylan2 282 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... M ) )  ->  ( F `  k )  =  A )
2221sumeq2dv 11077 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... M ) ( F `  k
)  =  sum_ k  e.  ( M ... M
) A )
23 fveq2 5387 . . . . . . 7  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2423eleq1d 2184 . . . . . 6  |-  ( k  =  M  ->  (
( F `  k
)  e.  CC  <->  ( F `  M )  e.  CC ) )
259, 10eqeltrd 2192 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2625ralrimiva 2480 . . . . . 6  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
275, 1syl6eleqr 2209 . . . . . 6  |-  ( ph  ->  M  e.  Z )
2824, 26, 27rspcdva 2766 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  CC )
2923fsum1 11121 . . . . 5  |-  ( ( M  e.  ZZ  /\  ( F `  M )  e.  CC )  ->  sum_ k  e.  ( M ... M ) ( F `  k )  =  ( F `  M ) )
303, 28, 29syl2anc 406 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... M ) ( F `  k
)  =  ( F `
 M ) )
3118, 22, 303eqtr2d 2154 . . 3  |-  ( ph  -> 
sum_ k  e.  ( M ... ( ( M  +  1 )  -  1 ) ) A  =  ( F `
 M ) )
3231oveq1d 5755 . 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 ) )
3312, 32eqtrd 2148 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 103    = wceq 1314    e. wcel 1463   dom cdm 4507   ` cfv 5091  (class class class)co 5740   CCcc 7582   1c1 7585    + caddc 7587    - cmin 7897   ZZcz 9005   ZZ>=cuz 9275   ...cfz 9730    seqcseq 10158    ~~> cli 10987   sum_csu 11062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-fz 9731  df-fzo 9860  df-seqfrec 10159  df-exp 10233  df-ihash 10462  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-clim 10988  df-sumdc 11063
This theorem is referenced by:  isumnn0nn  11202  efsep  11296
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