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Theorem gamcvg2lem 23714
Description: Lemma for gamcvg2 23715. (Contributed by Mario Carneiro, 10-Jul-2017.)
Hypotheses
Ref Expression
gamcvg2.f  |-  F  =  ( m  e.  NN  |->  ( ( ( ( m  +  1 )  /  m )  ^c  A )  /  (
( A  /  m
)  +  1 ) ) )
gamcvg2.a  |-  ( ph  ->  A  e.  ( CC 
\  ( ZZ  \  NN ) ) )
gamcvg2.g  |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( A  /  m )  +  1 ) ) ) )
Assertion
Ref Expression
gamcvg2lem  |-  ( ph  ->  ( exp  o.  seq 1 (  +  ,  G ) )  =  seq 1 (  x.  ,  F ) )
Distinct variable groups:    A, m    ph, m
Allowed substitution hints:    F( m)    G( m)

Proof of Theorem gamcvg2lem
Dummy variables  k  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcl 9604 . . . . 5  |-  ( ( n  e.  CC  /\  x  e.  CC )  ->  ( n  +  x
)  e.  CC )
21adantl 464 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
n  e.  CC  /\  x  e.  CC )
)  ->  ( n  +  x )  e.  CC )
3 simpll 752 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  e.  ( 1 ... k
) )  ->  ph )
4 elfznn 11768 . . . . . 6  |-  ( n  e.  ( 1 ... k )  ->  n  e.  NN )
54adantl 464 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  e.  ( 1 ... k
) )  ->  n  e.  NN )
6 oveq1 6285 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
m  +  1 )  =  ( n  + 
1 ) )
7 id 22 . . . . . . . . . . . 12  |-  ( m  =  n  ->  m  =  n )
86, 7oveq12d 6296 . . . . . . . . . . 11  |-  ( m  =  n  ->  (
( m  +  1 )  /  m )  =  ( ( n  +  1 )  /  n ) )
98fveq2d 5853 . . . . . . . . . 10  |-  ( m  =  n  ->  ( log `  ( ( m  +  1 )  /  m ) )  =  ( log `  (
( n  +  1 )  /  n ) ) )
109oveq2d 6294 . . . . . . . . 9  |-  ( m  =  n  ->  ( A  x.  ( log `  ( ( m  + 
1 )  /  m
) ) )  =  ( A  x.  ( log `  ( ( n  +  1 )  /  n ) ) ) )
11 oveq2 6286 . . . . . . . . . . 11  |-  ( m  =  n  ->  ( A  /  m )  =  ( A  /  n
) )
1211oveq1d 6293 . . . . . . . . . 10  |-  ( m  =  n  ->  (
( A  /  m
)  +  1 )  =  ( ( A  /  n )  +  1 ) )
1312fveq2d 5853 . . . . . . . . 9  |-  ( m  =  n  ->  ( log `  ( ( A  /  m )  +  1 ) )  =  ( log `  (
( A  /  n
)  +  1 ) ) )
1410, 13oveq12d 6296 . . . . . . . 8  |-  ( m  =  n  ->  (
( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
( A  /  m
)  +  1 ) ) )  =  ( ( A  x.  ( log `  ( ( n  +  1 )  /  n ) ) )  -  ( log `  (
( A  /  n
)  +  1 ) ) ) )
15 gamcvg2.g . . . . . . . 8  |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( A  /  m )  +  1 ) ) ) )
16 ovex 6306 . . . . . . . 8  |-  ( ( A  x.  ( log `  ( ( n  + 
1 )  /  n
) ) )  -  ( log `  ( ( A  /  n )  +  1 ) ) )  e.  _V
1714, 15, 16fvmpt 5932 . . . . . . 7  |-  ( n  e.  NN  ->  ( G `  n )  =  ( ( A  x.  ( log `  (
( n  +  1 )  /  n ) ) )  -  ( log `  ( ( A  /  n )  +  1 ) ) ) )
1817adantl 464 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  =  ( ( A  x.  ( log `  ( ( n  +  1 )  /  n ) ) )  -  ( log `  ( ( A  /  n )  +  1 ) ) ) )
19 gamcvg2.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( CC 
\  ( ZZ  \  NN ) ) )
2019adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )
2120eldifad 3426 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  A  e.  CC )
22 simpr 459 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  NN )
2322peano2nnd 10593 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( n  +  1 )  e.  NN )
2423nnrpd 11302 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( n  +  1 )  e.  RR+ )
2522nnrpd 11302 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  RR+ )
2624, 25rpdivcld 11321 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( n  +  1 )  /  n )  e.  RR+ )
2726relogcld 23302 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( log `  ( ( n  + 
1 )  /  n
) )  e.  RR )
2827recnd 9652 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( log `  ( ( n  + 
1 )  /  n
) )  e.  CC )
2921, 28mulcld 9646 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  ( log `  (
( n  +  1 )  /  n ) ) )  e.  CC )
3022nncnd 10592 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  CC )
3122nnne0d 10621 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  n  =/=  0 )
3221, 30, 31divcld 10361 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  /  n )  e.  CC )
33 1cnd 9642 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  1  e.  CC )
3432, 33addcld 9645 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( A  /  n )  +  1 )  e.  CC )
3520, 22dmgmdivn0 23683 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( A  /  n )  +  1 )  =/=  0 )
3634, 35logcld 23250 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( log `  ( ( A  /  n )  +  1 ) )  e.  CC )
3729, 36subcld 9967 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( A  x.  ( log `  ( ( n  + 
1 )  /  n
) ) )  -  ( log `  ( ( A  /  n )  +  1 ) ) )  e.  CC )
3818, 37eqeltrd 2490 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  CC )
393, 5, 38syl2anc 659 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  e.  ( 1 ... k
) )  ->  ( G `  n )  e.  CC )
40 simpr 459 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
41 nnuz 11162 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
4240, 41syl6eleq 2500 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  ( ZZ>= `  1 )
)
43 efadd 14038 . . . . 5  |-  ( ( n  e.  CC  /\  x  e.  CC )  ->  ( exp `  (
n  +  x ) )  =  ( ( exp `  n )  x.  ( exp `  x
) ) )
4443adantl 464 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
n  e.  CC  /\  x  e.  CC )
)  ->  ( exp `  ( n  +  x
) )  =  ( ( exp `  n
)  x.  ( exp `  x ) ) )
45 efsub 14044 . . . . . . . 8  |-  ( ( ( A  x.  ( log `  ( ( n  +  1 )  /  n ) ) )  e.  CC  /\  ( log `  ( ( A  /  n )  +  1 ) )  e.  CC )  ->  ( exp `  ( ( A  x.  ( log `  (
( n  +  1 )  /  n ) ) )  -  ( log `  ( ( A  /  n )  +  1 ) ) ) )  =  ( ( exp `  ( A  x.  ( log `  (
( n  +  1 )  /  n ) ) ) )  / 
( exp `  ( log `  ( ( A  /  n )  +  1 ) ) ) ) )
4629, 36, 45syl2anc 659 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( exp `  ( ( A  x.  ( log `  ( ( n  +  1 )  /  n ) ) )  -  ( log `  ( ( A  /  n )  +  1 ) ) ) )  =  ( ( exp `  ( A  x.  ( log `  ( ( n  +  1 )  /  n ) ) ) )  /  ( exp `  ( log `  (
( A  /  n
)  +  1 ) ) ) ) )
4730, 33addcld 9645 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( n  +  1 )  e.  CC )
4847, 30, 31divcld 10361 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( n  +  1 )  /  n )  e.  CC )
4923nnne0d 10621 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( n  +  1 )  =/=  0 )
5047, 30, 49, 31divne0d 10377 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( n  +  1 )  /  n )  =/=  0 )
5148, 50, 21cxpefd 23387 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( n  +  1 )  /  n )  ^c  A )  =  ( exp `  ( A  x.  ( log `  ( ( n  + 
1 )  /  n
) ) ) ) )
5251eqcomd 2410 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( exp `  ( A  x.  ( log `  ( ( n  +  1 )  /  n ) ) ) )  =  ( ( ( n  +  1 )  /  n )  ^c  A ) )
53 eflog 23256 . . . . . . . . 9  |-  ( ( ( ( A  /  n )  +  1 )  e.  CC  /\  ( ( A  /  n )  +  1 )  =/=  0 )  ->  ( exp `  ( log `  ( ( A  /  n )  +  1 ) ) )  =  ( ( A  /  n )  +  1 ) )
5434, 35, 53syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( exp `  ( log `  (
( A  /  n
)  +  1 ) ) )  =  ( ( A  /  n
)  +  1 ) )
5552, 54oveq12d 6296 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( exp `  ( A  x.  ( log `  (
( n  +  1 )  /  n ) ) ) )  / 
( exp `  ( log `  ( ( A  /  n )  +  1 ) ) ) )  =  ( ( ( ( n  + 
1 )  /  n
)  ^c  A )  /  ( ( A  /  n )  +  1 ) ) )
5646, 55eqtrd 2443 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( exp `  ( ( A  x.  ( log `  ( ( n  +  1 )  /  n ) ) )  -  ( log `  ( ( A  /  n )  +  1 ) ) ) )  =  ( ( ( ( n  +  1 )  /  n )  ^c  A )  /  ( ( A  /  n )  +  1 ) ) )
5718fveq2d 5853 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( exp `  ( G `  n
) )  =  ( exp `  ( ( A  x.  ( log `  ( ( n  + 
1 )  /  n
) ) )  -  ( log `  ( ( A  /  n )  +  1 ) ) ) ) )
588oveq1d 6293 . . . . . . . . 9  |-  ( m  =  n  ->  (
( ( m  + 
1 )  /  m
)  ^c  A )  =  ( ( ( n  +  1 )  /  n )  ^c  A ) )
5958, 12oveq12d 6296 . . . . . . . 8  |-  ( m  =  n  ->  (
( ( ( m  +  1 )  /  m )  ^c  A )  /  (
( A  /  m
)  +  1 ) )  =  ( ( ( ( n  + 
1 )  /  n
)  ^c  A )  /  ( ( A  /  n )  +  1 ) ) )
60 gamcvg2.f . . . . . . . 8  |-  F  =  ( m  e.  NN  |->  ( ( ( ( m  +  1 )  /  m )  ^c  A )  /  (
( A  /  m
)  +  1 ) ) )
61 ovex 6306 . . . . . . . 8  |-  ( ( ( ( n  + 
1 )  /  n
)  ^c  A )  /  ( ( A  /  n )  +  1 ) )  e.  _V
6259, 60, 61fvmpt 5932 . . . . . . 7  |-  ( n  e.  NN  ->  ( F `  n )  =  ( ( ( ( n  +  1 )  /  n )  ^c  A )  /  ( ( A  /  n )  +  1 ) ) )
6362adantl 464 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =  ( ( ( ( n  +  1 )  /  n )  ^c  A )  /  (
( A  /  n
)  +  1 ) ) )
6456, 57, 633eqtr4d 2453 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( exp `  ( G `  n
) )  =  ( F `  n ) )
653, 5, 64syl2anc 659 . . . 4  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  e.  ( 1 ... k
) )  ->  ( exp `  ( G `  n ) )  =  ( F `  n
) )
662, 39, 42, 44, 65seqhomo 12198 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( exp `  (  seq 1
(  +  ,  G
) `  k )
)  =  (  seq 1 (  x.  ,  F ) `  k
) )
6766mpteq2dva 4481 . 2  |-  ( ph  ->  ( k  e.  NN  |->  ( exp `  (  seq 1 (  +  ,  G ) `  k
) ) )  =  ( k  e.  NN  |->  (  seq 1 (  x.  ,  F ) `  k ) ) )
68 eff 14026 . . . 4  |-  exp : CC
--> CC
6968a1i 11 . . 3  |-  ( ph  ->  exp : CC --> CC )
70 1z 10935 . . . . 5  |-  1  e.  ZZ
7170a1i 11 . . . 4  |-  ( ph  ->  1  e.  ZZ )
7241, 71, 38serf 12179 . . 3  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> CC )
73 fcompt 6046 . . 3  |-  ( ( exp : CC --> CC  /\  seq 1 (  +  ,  G ) : NN --> CC )  ->  ( exp 
o.  seq 1 (  +  ,  G ) )  =  ( k  e.  NN  |->  ( exp `  (  seq 1 (  +  ,  G ) `  k
) ) ) )
7469, 72, 73syl2anc 659 . 2  |-  ( ph  ->  ( exp  o.  seq 1 (  +  ,  G ) )  =  ( k  e.  NN  |->  ( exp `  (  seq 1 (  +  ,  G ) `  k
) ) ) )
75 seqfn 12163 . . . . 5  |-  ( 1  e.  ZZ  ->  seq 1 (  x.  ,  F )  Fn  ( ZZ>=
`  1 ) )
7670, 75mp1i 13 . . . 4  |-  ( ph  ->  seq 1 (  x.  ,  F )  Fn  ( ZZ>= `  1 )
)
7741fneq2i 5657 . . . 4  |-  (  seq 1 (  x.  ,  F )  Fn  NN  <->  seq 1 (  x.  ,  F )  Fn  ( ZZ>=
`  1 ) )
7876, 77sylibr 212 . . 3  |-  ( ph  ->  seq 1 (  x.  ,  F )  Fn  NN )
79 dffn5 5894 . . 3  |-  (  seq 1 (  x.  ,  F )  Fn  NN  <->  seq 1 (  x.  ,  F )  =  ( k  e.  NN  |->  (  seq 1 (  x.  ,  F ) `  k ) ) )
8078, 79sylib 196 . 2  |-  ( ph  ->  seq 1 (  x.  ,  F )  =  ( k  e.  NN  |->  (  seq 1 (  x.  ,  F ) `  k ) ) )
8167, 74, 803eqtr4d 2453 1  |-  ( ph  ->  ( exp  o.  seq 1 (  +  ,  G ) )  =  seq 1 (  x.  ,  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598    \ cdif 3411    |-> cmpt 4453    o. ccom 4827    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   CCcc 9520   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527    - cmin 9841    / cdiv 10247   NNcn 10576   ZZcz 10905   ZZ>=cuz 11127   ...cfz 11726    seqcseq 12151   expce 14006   logclog 23234    ^c ccxp 23235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ioc 11587  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-ef 14012  df-sin 14014  df-cos 14015  df-pi 14017  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-haus 20109  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674  df-limc 22562  df-dv 22563  df-log 23236  df-cxp 23237
This theorem is referenced by:  gamcvg2  23715
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