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Theorem isum1p 11535
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 2189 . . 3  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  ( M  + 
1 ) )
3 isum1p.3 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
4 uzid 9573 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
53, 4syl 14 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
6 peano2uz 9615 . . . . 5  |-  ( M  e.  ( ZZ>= `  M
)  ->  ( M  +  1 )  e.  ( ZZ>= `  M )
)
75, 6syl 14 . . . 4  |-  ( ph  ->  ( M  +  1 )  e.  ( ZZ>= `  M ) )
87, 1eleqtrrdi 2283 . . 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 11534 . 2  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( sum_ k  e.  ( M ... ( ( M  + 
1 )  -  1 ) ) A  +  sum_ k  e.  ( ZZ>= `  ( M  +  1
) ) A ) )
133zcnd 9407 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
14 ax-1cn 7935 . . . . . . 7  |-  1  e.  CC
15 pncan 8194 . . . . . . 7  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( ( M  + 
1 )  -  1 )  =  M )
1613, 14, 15sylancl 413 . . . . . 6  |-  ( ph  ->  ( ( M  + 
1 )  -  1 )  =  M )
1716oveq2d 5913 . . . . 5  |-  ( ph  ->  ( M ... (
( M  +  1 )  -  1 ) )  =  ( M ... M ) )
1817sumeq1d 11409 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... ( ( M  +  1 )  -  1 ) ) A  =  sum_ k  e.  ( M ... M
) A )
19 elfzuz 10053 . . . . . . 7  |-  ( k  e.  ( M ... M )  ->  k  e.  ( ZZ>= `  M )
)
2019, 1eleqtrrdi 2283 . . . . . 6  |-  ( k  e.  ( M ... M )  ->  k  e.  Z )
2120, 9sylan2 286 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... M ) )  ->  ( F `  k )  =  A )
2221sumeq2dv 11411 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... M ) ( F `  k
)  =  sum_ k  e.  ( M ... M
) A )
23 fveq2 5534 . . . . . . 7  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2423eleq1d 2258 . . . . . 6  |-  ( k  =  M  ->  (
( F `  k
)  e.  CC  <->  ( F `  M )  e.  CC ) )
259, 10eqeltrd 2266 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2625ralrimiva 2563 . . . . . 6  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
275, 1eleqtrrdi 2283 . . . . . 6  |-  ( ph  ->  M  e.  Z )
2824, 26, 27rspcdva 2861 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  CC )
2923fsum1 11455 . . . . 5  |-  ( ( M  e.  ZZ  /\  ( F `  M )  e.  CC )  ->  sum_ k  e.  ( M ... M ) ( F `  k )  =  ( F `  M ) )
303, 28, 29syl2anc 411 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( M ... M ) ( F `  k
)  =  ( F `
 M ) )
3118, 22, 303eqtr2d 2228 . . 3  |-  ( ph  -> 
sum_ k  e.  ( M ... ( ( M  +  1 )  -  1 ) ) A  =  ( F `
 M ) )
3231oveq1d 5912 . 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 2222 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 104    = wceq 1364    e. wcel 2160   dom cdm 4644   ` cfv 5235  (class class class)co 5897   CCcc 7840   1c1 7843    + caddc 7845    - cmin 8159   ZZcz 9284   ZZ>=cuz 9559   ...cfz 10040    seqcseq 10478    ~~> cli 11321   sum_csu 11396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-frec 6417  df-1o 6442  df-oadd 6446  df-er 6560  df-en 6768  df-dom 6769  df-fin 6770  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-seqfrec 10479  df-exp 10554  df-ihash 10791  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-clim 11322  df-sumdc 11397
This theorem is referenced by:  isumnn0nn  11536  efsep  11734
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