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Theorem sge0sn 38009
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 4659 . . . . 5  |-  { A }  e.  _V
21a1i 11 . . . 4  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  { A }  e.  _V )
3 sge0sn.2 . . . . 5  |-  ( ph  ->  F : { A }
--> ( 0 [,] +oo ) )
43adantr 466 . . . 4  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  F : { A } --> ( 0 [,] +oo ) )
5 id 23 . . . . . . 7  |-  ( ( F `  A )  = +oo  ->  ( F `  A )  = +oo )
65eqcomd 2430 . . . . . 6  |-  ( ( F `  A )  = +oo  -> +oo  =  ( F `  A ) )
76adantl 467 . . . . 5  |-  ( (
ph  /\  ( F `  A )  = +oo )  -> +oo  =  ( F `  A )
)
8 ffun 5745 . . . . . . . 8  |-  ( F : { A } --> ( 0 [,] +oo )  ->  Fun  F )
93, 8syl 17 . . . . . . 7  |-  ( ph  ->  Fun  F )
109adantr 466 . . . . . 6  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  Fun  F )
11 sge0sn.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  V )
12 snidg 4022 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  { A } )
1311, 12syl 17 . . . . . . . 8  |-  ( ph  ->  A  e.  { A } )
14 fdm 5747 . . . . . . . . . 10  |-  ( F : { A } --> ( 0 [,] +oo )  ->  dom  F  =  { A } )
153, 14syl 17 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  { A } )
1615eqcomd 2430 . . . . . . . 8  |-  ( ph  ->  { A }  =  dom  F )
1713, 16eleqtrd 2512 . . . . . . 7  |-  ( ph  ->  A  e.  dom  F
)
1817adantr 466 . . . . . 6  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  A  e.  dom  F )
19 fvelrn 6027 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
2010, 18, 19syl2anc 665 . . . . 5  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  ( F `  A )  e.  ran  F )
217, 20eqeltrd 2510 . . . 4  |-  ( (
ph  /\  ( F `  A )  = +oo )  -> +oo  e.  ran  F )
222, 4, 21sge0pnfval 38003 . . 3  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  (Σ^ `  F )  = +oo )
23 simpr 462 . . 3  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  ( F `  A )  = +oo )
2422, 23eqtr4d 2466 . 2  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  (Σ^ `  F )  =  ( F `  A ) )
251a1i 11 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  { A }  e.  _V )
263adantr 466 . . . . 5  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  F : { A } --> ( 0 [,] +oo ) )
27 elsni 4021 . . . . . . . . 9  |-  ( +oo  e.  { ( F `  A ) }  -> +oo  =  ( F `  A ) )
2827eqcomd 2430 . . . . . . . 8  |-  ( +oo  e.  { ( F `  A ) }  ->  ( F `  A )  = +oo )
2928con3i 140 . . . . . . 7  |-  ( -.  ( F `  A
)  = +oo  ->  -. +oo  e.  { ( F `
 A ) } )
3029adantl 467 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  -. +oo  e.  { ( F `
 A ) } )
3111, 3rnsnf 37308 . . . . . . . 8  |-  ( ph  ->  ran  F  =  {
( F `  A
) } )
3231eqcomd 2430 . . . . . . 7  |-  ( ph  ->  { ( F `  A ) }  =  ran  F )
3332adantr 466 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  { ( F `  A ) }  =  ran  F
)
3430, 33neleqtrd 2534 . . . . 5  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  -. +oo  e.  ran  F )
3526, 34fge0iccico 38000 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  F : { A } --> ( 0 [,) +oo ) )
3625, 35sge0reval 38002 . . 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 13775 . . . . . . . 8  |-  sum_ y  e.  (/)  ( F `  y )  =  0
3837eqcomi 2435 . . . . . . 7  |-  0  =  sum_ y  e.  (/)  ( F `  y )
3938a1i 11 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  0  =  sum_ y  e.  (/)  ( F `  y ) )
40 nfcvd 2585 . . . . . . . 8  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  F/_ y
( F `  A
) )
41 nfv 1751 . . . . . . . 8  |-  F/ y ( ph  /\  -.  ( F `  A )  = +oo )
42 fveq2 5878 . . . . . . . . 9  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
4342adantl 467 . . . . . . . 8  |-  ( ( ( ph  /\  -.  ( F `  A )  = +oo )  /\  y  =  A )  ->  ( F `  y
)  =  ( F `
 A ) )
4411adantr 466 . . . . . . . 8  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  A  e.  V )
45 rge0ssre 11741 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
46 ax-resscn 9597 . . . . . . . . . 10  |-  RR  C_  CC
4745, 46sstri 3473 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  CC
4844, 12syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  A  e.  { A } )
4935, 48ffvelrnd 6035 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  e.  ( 0 [,) +oo ) )
5047, 49sseldi 3462 . . . . . . . 8  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  e.  CC )
5140, 41, 43, 44, 50sumsnd 37208 . . . . . . 7  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  sum_ y  e.  { A }  ( F `  y )  =  ( F `  A ) )
5251eqcomd 2430 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  =  sum_ y  e.  { A }  ( F `  y ) )
5339, 52preq12d 4084 . . . . 5  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  { 0 ,  ( F `  A ) }  =  { sum_ y  e.  (/)  ( F `  y ) ,  sum_ y  e.  { A }  ( F `  y ) } )
5453supeq1d 7963 . . . 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 11441 . . . . . . . 8  |-  <  Or  RR*
5655a1i 11 . . . . . . 7  |-  ( ph  ->  <  Or  RR* )
57 0xr 9688 . . . . . . . 8  |-  0  e.  RR*
5857a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  RR* )
59 iccssxr 11718 . . . . . . . 8  |-  ( 0 [,] +oo )  C_  RR*
603, 13ffvelrnd 6035 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  ( 0 [,] +oo ) )
6159, 60sseldi 3462 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR* )
62 suppr 7990 . . . . . . 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 1264 . . . . . 6  |-  ( ph  ->  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  )  =  if ( ( F `
 A )  <  0 ,  0 ,  ( F `  A
) ) )
64 pnfxr 11413 . . . . . . . . . . 11  |- +oo  e.  RR*
6564a1i 11 . . . . . . . . . 10  |-  ( ph  -> +oo  e.  RR* )
6658, 65, 603jca 1185 . . . . . . . . 9  |-  ( ph  ->  ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( F `  A )  e.  ( 0 [,] +oo ) ) )
67 iccgelb 11692 . . . . . . . . 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 9701 . . . . . . . 8  |-  ( ph  ->  ( 0  <_  ( F `  A )  <->  -.  ( F `  A
)  <  0 ) )
7068, 69mpbid 213 . . . . . . 7  |-  ( ph  ->  -.  ( F `  A )  <  0
)
7170iffalsed 3920 . . . . . 6  |-  ( ph  ->  if ( ( F `
 A )  <  0 ,  0 ,  ( F `  A
) )  =  ( F `  A ) )
7263, 71eqtr2d 2464 . . . . 5  |-  ( ph  ->  ( F `  A
)  =  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  ) )
7372adantr 466 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  =  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  ) )
74 pwsn 4210 . . . . . . . . . . . 12  |-  ~P { A }  =  { (/)
,  { A } }
7574ineq1i 3660 . . . . . . . . . . 11  |-  ( ~P { A }  i^i  Fin )  =  ( {
(/) ,  { A } }  i^i  Fin )
76 0fin 7802 . . . . . . . . . . . . 13  |-  (/)  e.  Fin
77 snfi 7654 . . . . . . . . . . . . 13  |-  { A }  e.  Fin
78 prssi 4153 . . . . . . . . . . . . 13  |-  ( (
(/)  e.  Fin  /\  { A }  e.  Fin )  ->  { (/) ,  { A } }  C_  Fin )
7976, 77, 78mp2an 676 . . . . . . . . . . . 12  |-  { (/) ,  { A } }  C_ 
Fin
80 df-ss 3450 . . . . . . . . . . . . 13  |-  ( {
(/) ,  { A } }  C_  Fin  <->  ( { (/)
,  { A } }  i^i  Fin )  =  { (/) ,  { A } } )
8180biimpi 197 . . . . . . . . . . . 12  |-  ( {
(/) ,  { A } }  C_  Fin  ->  ( { (/) ,  { A } }  i^i  Fin )  =  { (/) ,  { A } } )
8279, 81ax-mp 5 . . . . . . . . . . 11  |-  ( {
(/) ,  { A } }  i^i  Fin )  =  { (/) ,  { A } }
8375, 82eqtri 2451 . . . . . . . . . 10  |-  ( ~P { A }  i^i  Fin )  =  { (/) ,  { A } }
84 mpteq1 4501 . . . . . . . . . 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 4553 . . . . . . . . . . . . 13  |-  (/)  e.  _V
8786a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  (/)  e.  _V )
881a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  { A }  e.  _V )
89 sumex 13742 . . . . . . . . . . . . 13  |-  sum_ y  e.  (/)  ( F `  y )  e.  _V
9089a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  sum_ y  e.  (/)  ( F `  y )  e.  _V )
91 sumex 13742 . . . . . . . . . . . . 13  |-  sum_ y  e.  { A }  ( F `  y )  e.  _V
9291a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  sum_ y  e.  { A }  ( F `  y )  e.  _V )
93 sumeq1 13743 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  (/)  ( F `
 y ) )
9493adantl 467 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  =  (/) )  ->  sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  (/)  ( F `
 y ) )
95 sumeq1 13743 . . . . . . . . . . . . 13  |-  ( x  =  { A }  -> 
sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  { A }  ( F `  y ) )
9695adantl 467 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  =  { A } )  ->  sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  { A }  ( F `  y ) )
9787, 88, 90, 92, 94, 96fmptpr 6101 . . . . . . . . . . 11  |-  ( T. 
->  { <. (/) ,  sum_ y  e.  (/)  ( F `  y ) >. ,  <. { A } ,  sum_ y  e.  { A }  ( F `  y ) >. }  =  ( x  e.  { (/) ,  { A } }  |-> 
sum_ y  e.  x  ( F `  y ) ) )
9897trud 1446 . . . . . . . . . 10  |-  { <. (/)
,  sum_ y  e.  (/)  ( F `  y )
>. ,  <. { A } ,  sum_ y  e. 
{ A }  ( F `  y ) >. }  =  ( x  e.  { (/) ,  { A } }  |->  sum_ y  e.  x  ( F `  y ) )
9998eqcomi 2435 . . . . . . . . 9  |-  ( x  e.  { (/) ,  { A } }  |->  sum_ y  e.  x  ( F `  y ) )  =  { <. (/) ,  sum_ y  e.  (/)  ( F `  y ) >. ,  <. { A } ,  sum_ y  e.  { A }  ( F `  y ) >. }
10085, 99eqtri 2451 . . . . . . . 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 5077 . . . . . . 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 5332 . . . . . . . 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 676 . . . . . . 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 2451 . . . . . 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 7964 . . . . 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 2473 . . 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 2466 . 2  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  (Σ^ `  F
)  =  ( F `
 A ) )
10924, 108pm2.61dan 798 1  |-  ( ph  ->  (Σ^ `  F )  =  ( F `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   T. wtru 1438    e. wcel 1868   _Vcvv 3081    i^i cin 3435    C_ wss 3436   (/)c0 3761   ifcif 3909   ~Pcpw 3979   {csn 3996   {cpr 3998   <.cop 4002   class class class wbr 4420    |-> cmpt 4479    Or wor 4770   dom cdm 4850   ran crn 4851   Fun wfun 5592   -->wf 5594   ` cfv 5598  (class class class)co 6302   Fincfn 7574   supcsup 7957   CCcc 9538   RRcr 9539   0cc0 9540   +oocpnf 9673   RR*cxr 9675    < clt 9676    <_ cle 9677   [,)cico 11638   [,]cicc 11639   sum_csu 13740  Σ^csumge0 37992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-sumge0 37993
This theorem is referenced by:  sge0snmpt  38013  sge0sup  38021  caratheodorylem1  38126
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