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Theorem isumsplit 11211
Description: Split off the first  N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 21-Oct-2022.)
Hypotheses
Ref Expression
isumsplit.1  |-  Z  =  ( ZZ>= `  M )
isumsplit.2  |-  W  =  ( ZZ>= `  N )
isumsplit.3  |-  ( ph  ->  N  e.  Z )
isumsplit.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
isumsplit.5  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
isumsplit.6  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
Assertion
Ref Expression
isumsplit  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
Distinct variable groups:    k, F    k, M    ph, k    k, Z   
k, N    k, W
Allowed substitution hint:    A( k)

Proof of Theorem isumsplit
Dummy variables  j  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isumsplit.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 isumsplit.3 . . . 4  |-  ( ph  ->  N  e.  Z )
32, 1syl6eleq 2208 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 eluzel2 9283 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
53, 4syl 14 . 2  |-  ( ph  ->  M  e.  ZZ )
6 isumsplit.4 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
7 isumsplit.5 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
8 isumsplit.2 . . 3  |-  W  =  ( ZZ>= `  N )
9 eluzelz 9287 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
103, 9syl 14 . . 3  |-  ( ph  ->  N  e.  ZZ )
11 uzss 9298 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  M ) )
123, 11syl 14 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
)
1312, 8, 13sstr4g 3108 . . . . . 6  |-  ( ph  ->  W  C_  Z )
1413sselda 3065 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  k  e.  Z )
1514, 6syldan 278 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  =  A )
1614, 7syldan 278 . . . 4  |-  ( (
ph  /\  k  e.  W )  ->  A  e.  CC )
17 isumsplit.6 . . . . 5  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
186, 7eqeltrd 2192 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
191, 2, 18iserex 11059 . . . . 5  |-  ( ph  ->  (  seq M (  +  ,  F )  e.  dom  ~~>  <->  seq N (  +  ,  F )  e.  dom  ~~>  ) )
2017, 19mpbid 146 . . . 4  |-  ( ph  ->  seq N (  +  ,  F )  e. 
dom 
~~>  )
218, 10, 15, 16, 20isumclim2 11142 . . 3  |-  ( ph  ->  seq N (  +  ,  F )  ~~>  sum_ k  e.  W  A )
22 peano2zm 9046 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
2310, 22syl 14 . . . . 5  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
245, 23fzfigd 10155 . . . 4  |-  ( ph  ->  ( M ... ( N  -  1 ) )  e.  Fin )
25 elfzuz 9753 . . . . . 6  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  ( ZZ>= `  M )
)
2625, 1syl6eleqr 2209 . . . . 5  |-  ( k  e.  ( M ... ( N  -  1
) )  ->  k  e.  Z )
2726, 7sylan2 282 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  A  e.  CC )
2824, 27fsumcl 11120 . . 3  |-  ( ph  -> 
sum_ k  e.  ( M ... ( N  -  1 ) ) A  e.  CC )
2914, 18syldan 278 . . . . 5  |-  ( (
ph  /\  k  e.  W )  ->  ( F `  k )  e.  CC )
308, 10, 29serf 10198 . . . 4  |-  ( ph  ->  seq N (  +  ,  F ) : W --> CC )
3130ffvelrnda 5521 . . 3  |-  ( (
ph  /\  j  e.  W )  ->  (  seq N (  +  ,  F ) `  j
)  e.  CC )
325zred 9127 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  RR )
3332ltm1d 8650 . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  1 )  <  M )
34 peano2zm 9046 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
355, 34syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
36 fzn 9773 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
375, 35, 36syl2anc 406 . . . . . . . . . . 11  |-  ( ph  ->  ( ( M  - 
1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
3833, 37mpbid 146 . . . . . . . . . 10  |-  ( ph  ->  ( M ... ( M  -  1 ) )  =  (/) )
3938sumeq1d 11086 . . . . . . . . 9  |-  ( ph  -> 
sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  sum_ k  e.  (/)  A )
4039adantr 272 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  sum_ k  e.  (/)  A )
41 sum0 11108 . . . . . . . 8  |-  sum_ k  e.  (/)  A  =  0
4240, 41syl6eq 2164 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  sum_ k  e.  ( M ... ( M  -  1 ) ) A  =  0 )
4342oveq1d 5755 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  ( sum_ k  e.  ( M ... ( M  - 
1 ) ) A  +  (  seq M
(  +  ,  F
) `  j )
)  =  ( 0  +  (  seq M
(  +  ,  F
) `  j )
) )
4413sselda 3065 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  j  e.  Z )
451, 5, 18serf 10198 . . . . . . . . 9  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
4645ffvelrnda 5521 . . . . . . . 8  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ) `  j
)  e.  CC )
4744, 46syldan 278 . . . . . . 7  |-  ( (
ph  /\  j  e.  W )  ->  (  seq M (  +  ,  F ) `  j
)  e.  CC )
4847addid2d 7876 . . . . . 6  |-  ( (
ph  /\  j  e.  W )  ->  (
0  +  (  seq M (  +  ,  F ) `  j
) )  =  (  seq M (  +  ,  F ) `  j ) )
4943, 48eqtr2d 2149 . . . . 5  |-  ( (
ph  /\  j  e.  W )  ->  (  seq M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( M  -  1 ) ) A  +  (  seq M (  +  ,  F ) `  j ) ) )
50 oveq1 5747 . . . . . . . . 9  |-  ( N  =  M  ->  ( N  -  1 )  =  ( M  - 
1 ) )
5150oveq2d 5756 . . . . . . . 8  |-  ( N  =  M  ->  ( M ... ( N  - 
1 ) )  =  ( M ... ( M  -  1 ) ) )
5251sumeq1d 11086 . . . . . . 7  |-  ( N  =  M  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  sum_ k  e.  ( M ... ( M  -  1 ) ) A )
53 seqeq1 10172 . . . . . . . 8  |-  ( N  =  M  ->  seq N (  +  ,  F )  =  seq M (  +  ,  F ) )
5453fveq1d 5389 . . . . . . 7  |-  ( N  =  M  ->  (  seq N (  +  ,  F ) `  j
)  =  (  seq M (  +  ,  F ) `  j
) )
5552, 54oveq12d 5758 . . . . . 6  |-  ( N  =  M  ->  ( sum_ k  e.  ( M ... ( N  - 
1 ) ) A  +  (  seq N
(  +  ,  F
) `  j )
)  =  ( sum_ k  e.  ( M ... ( M  -  1 ) ) A  +  (  seq M (  +  ,  F ) `  j ) ) )
5655eqeq2d 2127 . . . . 5  |-  ( N  =  M  ->  (
(  seq M (  +  ,  F ) `  j )  =  (
sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq N (  +  ,  F ) `  j
) )  <->  (  seq M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( M  -  1 ) ) A  +  (  seq M (  +  ,  F ) `  j ) ) ) )
5749, 56syl5ibrcom 156 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  =  M  ->  (  seq M (  +  ,  F ) `  j )  =  (
sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq N (  +  ,  F ) `  j
) ) ) )
58 addcl 7709 . . . . . . . 8  |-  ( ( k  e.  CC  /\  m  e.  CC )  ->  ( k  +  m
)  e.  CC )
5958adantl 273 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  ( k  e.  CC  /\  m  e.  CC ) )  -> 
( k  +  m
)  e.  CC )
60 addass 7714 . . . . . . . 8  |-  ( ( k  e.  CC  /\  m  e.  CC  /\  x  e.  CC )  ->  (
( k  +  m
)  +  x )  =  ( k  +  ( m  +  x
) ) )
6160adantl 273 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  ( k  e.  CC  /\  m  e.  CC  /\  x  e.  CC ) )  -> 
( ( k  +  m )  +  x
)  =  ( k  +  ( m  +  x ) ) )
62 simplr 502 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  j  e.  W )
63 simpll 501 . . . . . . . . . . 11  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ph )
6410zcnd 9128 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
65 ax-1cn 7677 . . . . . . . . . . . . 13  |-  1  e.  CC
66 npcan 7935 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
6764, 65, 66sylancl 407 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
6867eqcomd 2121 . . . . . . . . . . 11  |-  ( ph  ->  N  =  ( ( N  -  1 )  +  1 ) )
6963, 68syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  N  =  ( ( N  - 
1 )  +  1 ) )
7069fveq2d 5391 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( ZZ>= `  N )  =  (
ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
718, 70syl5eq 2160 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  W  =  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
7262, 71eleqtrd 2194 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  j  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
735adantr 272 . . . . . . . 8  |-  ( (
ph  /\  j  e.  W )  ->  M  e.  ZZ )
74 eluzp1m1 9301 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( N  -  1 )  e.  ( ZZ>= `  M ) )
7573, 74sylan 279 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  M )
)
761eleq2i 2182 . . . . . . . . . 10  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
7776, 6sylan2br 284 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  A )
7863, 77sylan 279 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  (
ZZ>= `  M ) )  ->  ( F `  k )  =  A )
7976, 7sylan2br 284 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  A  e.  CC )
8063, 79sylan 279 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  (
ZZ>= `  M ) )  ->  A  e.  CC )
8178, 80eqeltrd 2192 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>=
`  ( M  + 
1 ) ) )  /\  k  e.  (
ZZ>= `  M ) )  ->  ( F `  k )  e.  CC )
8259, 61, 72, 75, 81seq3split 10203 . . . . . 6  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  +  ,  F ) `  j
)  =  ( (  seq M (  +  ,  F ) `  ( N  -  1
) )  +  (  seq ( ( N  -  1 )  +  1 ) (  +  ,  F ) `  j ) ) )
8378, 75, 80fsum3ser 11117 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  sum_ k  e.  ( M ... ( N  -  1 ) ) A  =  (  seq M (  +  ,  F ) `  ( N  -  1
) ) )
8469seqeq1d 10175 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  seq N (  +  ,  F )  =  seq ( ( N  -  1 )  +  1 ) (  +  ,  F ) )
8584fveq1d 5389 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq N (  +  ,  F ) `  j
)  =  (  seq ( ( N  - 
1 )  +  1 ) (  +  ,  F ) `  j
) )
8683, 85oveq12d 5758 . . . . . 6  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq N (  +  ,  F ) `  j ) )  =  ( (  seq M
(  +  ,  F
) `  ( N  -  1 ) )  +  (  seq (
( N  -  1 )  +  1 ) (  +  ,  F
) `  j )
) )
8782, 86eqtr4d 2151 . . . . 5  |-  ( ( ( ph  /\  j  e.  W )  /\  N  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq N (  +  ,  F ) `  j ) ) )
8887ex 114 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  e.  ( ZZ>= `  ( M  +  1
) )  ->  (  seq M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq N (  +  ,  F ) `  j ) ) ) )
89 uzp1 9311 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
903, 89syl 14 . . . . 5  |-  ( ph  ->  ( N  =  M  \/  N  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
9190adantr 272 . . . 4  |-  ( (
ph  /\  j  e.  W )  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1
) ) ) )
9257, 88, 91mpjaod 690 . . 3  |-  ( (
ph  /\  j  e.  W )  ->  (  seq M (  +  ,  F ) `  j
)  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  (  seq N (  +  ,  F ) `  j ) ) )
938, 10, 21, 28, 17, 31, 92climaddc2 11050 . 2  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
941, 5, 6, 7, 93isumclim 11141 1  |-  ( ph  -> 
sum_ k  e.  Z  A  =  ( sum_ k  e.  ( M ... ( N  -  1 ) ) A  +  sum_ k  e.  W  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463    C_ wss 3039   (/)c0 3331   class class class wbr 3897   dom cdm 4507   ` cfv 5091  (class class class)co 5740   CCcc 7582   0cc0 7584   1c1 7585    + caddc 7587    < clt 7764    - cmin 7897   ZZcz 9008   ZZ>=cuz 9278   ...cfz 9741    seqcseq 10169    ~~> cli 10998   sum_csu 11073
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 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fz 9742  df-fzo 9871  df-seqfrec 10170  df-exp 10244  df-ihash 10473  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-clim 10999  df-sumdc 11074
This theorem is referenced by:  isum1p  11212  geolim2  11232  mertenslem2  11256  mertensabs  11257  effsumlt  11308  eirraplem  11390  trilpolemeq1  13067  trilpolemlt1  13068
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