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Theorem sge0sn 38221
Description: A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0sn.1  |-  ( ph  ->  A  e.  V )
sge0sn.2  |-  ( ph  ->  F : { A }
--> ( 0 [,] +oo ) )
Assertion
Ref Expression
sge0sn  |-  ( ph  ->  (Σ^ `  F )  =  ( F `  A ) )

Proof of Theorem sge0sn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4641 . . . . 5  |-  { A }  e.  _V
21a1i 11 . . . 4  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  { A }  e.  _V )
3 sge0sn.2 . . . . 5  |-  ( ph  ->  F : { A }
--> ( 0 [,] +oo ) )
43adantr 467 . . . 4  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  F : { A } --> ( 0 [,] +oo ) )
5 id 22 . . . . . . 7  |-  ( ( F `  A )  = +oo  ->  ( F `  A )  = +oo )
65eqcomd 2457 . . . . . 6  |-  ( ( F `  A )  = +oo  -> +oo  =  ( F `  A ) )
76adantl 468 . . . . 5  |-  ( (
ph  /\  ( F `  A )  = +oo )  -> +oo  =  ( F `  A )
)
8 ffun 5731 . . . . . . . 8  |-  ( F : { A } --> ( 0 [,] +oo )  ->  Fun  F )
93, 8syl 17 . . . . . . 7  |-  ( ph  ->  Fun  F )
109adantr 467 . . . . . 6  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  Fun  F )
11 sge0sn.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  V )
12 snidg 3994 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  { A } )
1311, 12syl 17 . . . . . . . 8  |-  ( ph  ->  A  e.  { A } )
14 fdm 5733 . . . . . . . . . 10  |-  ( F : { A } --> ( 0 [,] +oo )  ->  dom  F  =  { A } )
153, 14syl 17 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  { A } )
1615eqcomd 2457 . . . . . . . 8  |-  ( ph  ->  { A }  =  dom  F )
1713, 16eleqtrd 2531 . . . . . . 7  |-  ( ph  ->  A  e.  dom  F
)
1817adantr 467 . . . . . 6  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  A  e.  dom  F )
19 fvelrn 6015 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
2010, 18, 19syl2anc 667 . . . . 5  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  ( F `  A )  e.  ran  F )
217, 20eqeltrd 2529 . . . 4  |-  ( (
ph  /\  ( F `  A )  = +oo )  -> +oo  e.  ran  F )
222, 4, 21sge0pnfval 38215 . . 3  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  (Σ^ `  F )  = +oo )
23 simpr 463 . . 3  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  ( F `  A )  = +oo )
2422, 23eqtr4d 2488 . 2  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  (Σ^ `  F )  =  ( F `  A ) )
251a1i 11 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  { A }  e.  _V )
263adantr 467 . . . . 5  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  F : { A } --> ( 0 [,] +oo ) )
27 elsni 3993 . . . . . . . . 9  |-  ( +oo  e.  { ( F `  A ) }  -> +oo  =  ( F `  A ) )
2827eqcomd 2457 . . . . . . . 8  |-  ( +oo  e.  { ( F `  A ) }  ->  ( F `  A )  = +oo )
2928con3i 141 . . . . . . 7  |-  ( -.  ( F `  A
)  = +oo  ->  -. +oo  e.  { ( F `
 A ) } )
3029adantl 468 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  -. +oo  e.  { ( F `
 A ) } )
3111, 3rnsnf 37458 . . . . . . . 8  |-  ( ph  ->  ran  F  =  {
( F `  A
) } )
3231eqcomd 2457 . . . . . . 7  |-  ( ph  ->  { ( F `  A ) }  =  ran  F )
3332adantr 467 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  { ( F `  A ) }  =  ran  F
)
3430, 33neleqtrd 2550 . . . . 5  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  -. +oo  e.  ran  F )
3526, 34fge0iccico 38212 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  F : { A } --> ( 0 [,) +oo ) )
3625, 35sge0reval 38214 . . 3  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  (Σ^ `  F
)  =  sup ( ran  ( x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) , 
RR* ,  <  ) )
37 sum0 13787 . . . . . . . 8  |-  sum_ y  e.  (/)  ( F `  y )  =  0
3837eqcomi 2460 . . . . . . 7  |-  0  =  sum_ y  e.  (/)  ( F `  y )
3938a1i 11 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  0  =  sum_ y  e.  (/)  ( F `  y ) )
40 nfcvd 2593 . . . . . . . 8  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  F/_ y
( F `  A
) )
41 nfv 1761 . . . . . . . 8  |-  F/ y ( ph  /\  -.  ( F `  A )  = +oo )
42 fveq2 5865 . . . . . . . . 9  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
4342adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  -.  ( F `  A )  = +oo )  /\  y  =  A )  ->  ( F `  y
)  =  ( F `
 A ) )
4411adantr 467 . . . . . . . 8  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  A  e.  V )
45 rge0ssre 11740 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
46 ax-resscn 9596 . . . . . . . . . 10  |-  RR  C_  CC
4745, 46sstri 3441 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  CC
4844, 12syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  A  e.  { A } )
4935, 48ffvelrnd 6023 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  e.  ( 0 [,) +oo ) )
5047, 49sseldi 3430 . . . . . . . 8  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  e.  CC )
5140, 41, 43, 44, 50sumsnd 37347 . . . . . . 7  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  sum_ y  e.  { A }  ( F `  y )  =  ( F `  A ) )
5251eqcomd 2457 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  =  sum_ y  e.  { A }  ( F `  y ) )
5339, 52preq12d 4059 . . . . 5  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  { 0 ,  ( F `  A ) }  =  { sum_ y  e.  (/)  ( F `  y ) ,  sum_ y  e.  { A }  ( F `  y ) } )
5453supeq1d 7960 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  )  =  sup ( {
sum_ y  e.  (/)  ( F `  y ) ,  sum_ y  e.  { A }  ( F `  y ) } ,  RR* ,  <  ) )
55 xrltso 11440 . . . . . . . 8  |-  <  Or  RR*
5655a1i 11 . . . . . . 7  |-  ( ph  ->  <  Or  RR* )
57 0xr 9687 . . . . . . . 8  |-  0  e.  RR*
5857a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  RR* )
59 iccssxr 11717 . . . . . . . 8  |-  ( 0 [,] +oo )  C_  RR*
603, 13ffvelrnd 6023 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  ( 0 [,] +oo ) )
6159, 60sseldi 3430 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR* )
62 suppr 7987 . . . . . . 7  |-  ( (  <  Or  RR*  /\  0  e.  RR*  /\  ( F `
 A )  e. 
RR* )  ->  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  )  =  if ( ( F `  A )  <  0 ,  0 ,  ( F `  A ) ) )
6356, 58, 61, 62syl3anc 1268 . . . . . 6  |-  ( ph  ->  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  )  =  if ( ( F `
 A )  <  0 ,  0 ,  ( F `  A
) ) )
64 pnfxr 11412 . . . . . . . . . . 11  |- +oo  e.  RR*
6564a1i 11 . . . . . . . . . 10  |-  ( ph  -> +oo  e.  RR* )
6658, 65, 603jca 1188 . . . . . . . . 9  |-  ( ph  ->  ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( F `  A )  e.  ( 0 [,] +oo ) ) )
67 iccgelb 11691 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( F `
 A )  e.  ( 0 [,] +oo ) )  ->  0  <_  ( F `  A
) )
6866, 67syl 17 . . . . . . . 8  |-  ( ph  ->  0  <_  ( F `  A ) )
6958, 61xrlenltd 9700 . . . . . . . 8  |-  ( ph  ->  ( 0  <_  ( F `  A )  <->  -.  ( F `  A
)  <  0 ) )
7068, 69mpbid 214 . . . . . . 7  |-  ( ph  ->  -.  ( F `  A )  <  0
)
7170iffalsed 3892 . . . . . 6  |-  ( ph  ->  if ( ( F `
 A )  <  0 ,  0 ,  ( F `  A
) )  =  ( F `  A ) )
7263, 71eqtr2d 2486 . . . . 5  |-  ( ph  ->  ( F `  A
)  =  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  ) )
7372adantr 467 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  =  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  ) )
74 pwsn 4192 . . . . . . . . . . . 12  |-  ~P { A }  =  { (/)
,  { A } }
7574ineq1i 3630 . . . . . . . . . . 11  |-  ( ~P { A }  i^i  Fin )  =  ( {
(/) ,  { A } }  i^i  Fin )
76 0fin 7799 . . . . . . . . . . . . 13  |-  (/)  e.  Fin
77 snfi 7650 . . . . . . . . . . . . 13  |-  { A }  e.  Fin
78 prssi 4128 . . . . . . . . . . . . 13  |-  ( (
(/)  e.  Fin  /\  { A }  e.  Fin )  ->  { (/) ,  { A } }  C_  Fin )
7976, 77, 78mp2an 678 . . . . . . . . . . . 12  |-  { (/) ,  { A } }  C_ 
Fin
80 df-ss 3418 . . . . . . . . . . . . 13  |-  ( {
(/) ,  { A } }  C_  Fin  <->  ( { (/)
,  { A } }  i^i  Fin )  =  { (/) ,  { A } } )
8180biimpi 198 . . . . . . . . . . . 12  |-  ( {
(/) ,  { A } }  C_  Fin  ->  ( { (/) ,  { A } }  i^i  Fin )  =  { (/) ,  { A } } )
8279, 81ax-mp 5 . . . . . . . . . . 11  |-  ( {
(/) ,  { A } }  i^i  Fin )  =  { (/) ,  { A } }
8375, 82eqtri 2473 . . . . . . . . . 10  |-  ( ~P { A }  i^i  Fin )  =  { (/) ,  { A } }
84 mpteq1 4483 . . . . . . . . . 10  |-  ( ( ~P { A }  i^i  Fin )  =  { (/)
,  { A } }  ->  ( x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  =  ( x  e.  { (/) ,  { A } }  |->  sum_ y  e.  x  ( F `  y ) ) )
8583, 84ax-mp 5 . . . . . . . . 9  |-  ( x  e.  ( ~P { A }  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  =  ( x  e.  { (/) ,  { A } }  |->  sum_ y  e.  x  ( F `  y ) )
86 0ex 4535 . . . . . . . . . . . . 13  |-  (/)  e.  _V
8786a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  (/)  e.  _V )
881a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  { A }  e.  _V )
89 sumex 13754 . . . . . . . . . . . . 13  |-  sum_ y  e.  (/)  ( F `  y )  e.  _V
9089a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  sum_ y  e.  (/)  ( F `  y )  e.  _V )
91 sumex 13754 . . . . . . . . . . . . 13  |-  sum_ y  e.  { A }  ( F `  y )  e.  _V
9291a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  sum_ y  e.  { A }  ( F `  y )  e.  _V )
93 sumeq1 13755 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  (/)  ( F `
 y ) )
9493adantl 468 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  =  (/) )  ->  sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  (/)  ( F `
 y ) )
95 sumeq1 13755 . . . . . . . . . . . . 13  |-  ( x  =  { A }  -> 
sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  { A }  ( F `  y ) )
9695adantl 468 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  =  { A } )  ->  sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  { A }  ( F `  y ) )
9787, 88, 90, 92, 94, 96fmptpr 6089 . . . . . . . . . . 11  |-  ( T. 
->  { <. (/) ,  sum_ y  e.  (/)  ( F `  y ) >. ,  <. { A } ,  sum_ y  e.  { A }  ( F `  y ) >. }  =  ( x  e.  { (/) ,  { A } }  |-> 
sum_ y  e.  x  ( F `  y ) ) )
9897trud 1453 . . . . . . . . . 10  |-  { <. (/)
,  sum_ y  e.  (/)  ( F `  y )
>. ,  <. { A } ,  sum_ y  e. 
{ A }  ( F `  y ) >. }  =  ( x  e.  { (/) ,  { A } }  |->  sum_ y  e.  x  ( F `  y ) )
9998eqcomi 2460 . . . . . . . . 9  |-  ( x  e.  { (/) ,  { A } }  |->  sum_ y  e.  x  ( F `  y ) )  =  { <. (/) ,  sum_ y  e.  (/)  ( F `  y ) >. ,  <. { A } ,  sum_ y  e.  { A }  ( F `  y ) >. }
10085, 99eqtri 2473 . . . . . . . 8  |-  ( x  e.  ( ~P { A }  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  =  { <. (/)
,  sum_ y  e.  (/)  ( F `  y )
>. ,  <. { A } ,  sum_ y  e. 
{ A }  ( F `  y ) >. }
101100rneqi 5061 . . . . . . 7  |-  ran  (
x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  =  ran  { <.
(/) ,  sum_ y  e.  (/)  ( F `  y
) >. ,  <. { A } ,  sum_ y  e. 
{ A }  ( F `  y ) >. }
102 rnpropg 5316 . . . . . . . 8  |-  ( (
(/)  e.  _V  /\  { A }  e.  _V )  ->  ran  { <. (/) ,  sum_ y  e.  (/)  ( F `
 y ) >. ,  <. { A } ,  sum_ y  e.  { A }  ( F `  y ) >. }  =  { sum_ y  e.  (/)  ( F `  y ) ,  sum_ y  e.  { A }  ( F `  y ) } )
10386, 1, 102mp2an 678 . . . . . . 7  |-  ran  { <.
(/) ,  sum_ y  e.  (/)  ( F `  y
) >. ,  <. { A } ,  sum_ y  e. 
{ A }  ( F `  y ) >. }  =  { sum_ y  e.  (/)  ( F `
 y ) , 
sum_ y  e.  { A }  ( F `  y ) }
104101, 103eqtri 2473 . . . . . 6  |-  ran  (
x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  =  { sum_ y  e.  (/)  ( F `
 y ) , 
sum_ y  e.  { A }  ( F `  y ) }
105104supeq1i 7961 . . . . 5  |-  sup ( ran  ( x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) , 
RR* ,  <  )  =  sup ( { sum_ y  e.  (/)  ( F `
 y ) , 
sum_ y  e.  { A }  ( F `  y ) } ,  RR* ,  <  )
106105a1i 11 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  sup ( ran  ( x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  )  =  sup ( {
sum_ y  e.  (/)  ( F `  y ) ,  sum_ y  e.  { A }  ( F `  y ) } ,  RR* ,  <  ) )
10754, 73, 1063eqtr4d 2495 . . 3  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  =  sup ( ran  (
x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  ) )
10836, 107eqtr4d 2488 . 2  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  (Σ^ `  F
)  =  ( F `
 A ) )
10924, 108pm2.61dan 800 1  |-  ( ph  ->  (Σ^ `  F )  =  ( F `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444   T. wtru 1445    e. wcel 1887   _Vcvv 3045    i^i cin 3403    C_ wss 3404   (/)c0 3731   ifcif 3881   ~Pcpw 3951   {csn 3968   {cpr 3970   <.cop 3974   class class class wbr 4402    |-> cmpt 4461    Or wor 4754   dom cdm 4834   ran crn 4835   Fun wfun 5576   -->wf 5578   ` cfv 5582  (class class class)co 6290   Fincfn 7569   supcsup 7954   CCcc 9537   RRcr 9538   0cc0 9539   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676   [,)cico 11637   [,]cicc 11638   sum_csu 13752  Σ^csumge0 38204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-sumge0 38205
This theorem is referenced by:  sge0snmpt  38225  sge0sup  38233  sge0snmptf  38279  caratheodorylem1  38347
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