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Theorem psrbasOLD 18010
Description: The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) Obsolete version of psrbas 18009 as of 8-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
psrbas.s  |-  S  =  ( I mPwSer  R )
psrbas.k  |-  K  =  ( Base `  R
)
psrbas.d  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
psrbas.b  |-  B  =  ( Base `  S
)
psrbas.i  |-  ( ph  ->  I  e.  V )
Assertion
Ref Expression
psrbasOLD  |-  ( ph  ->  B  =  ( K  ^m  D ) )
Distinct variable group:    f, I
Allowed substitution hints:    ph( f)    B( f)    D( f)    R( f)    S( f)    K( f)    V( f)

Proof of Theorem psrbasOLD
Dummy variables  g  h  k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrbas.s . . . . 5  |-  S  =  ( I mPwSer  R )
2 psrbas.k . . . . 5  |-  K  =  ( Base `  R
)
3 eqid 2443 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2443 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2443 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
6 psrbas.d . . . . 5  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
7 eqidd 2444 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  ( K  ^m  D )  =  ( K  ^m  D
) )
8 eqid 2443 . . . . 5  |-  (  oF ( +g  `  R
)  |`  ( ( K  ^m  D )  X.  ( K  ^m  D
) ) )  =  (  oF ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) )
9 eqid 2443 . . . . 5  |-  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D
)  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  oR  <_  k }  |->  ( ( g `  x ) ( .r
`  R ) ( h `  ( k  oF  -  x
) ) ) ) ) ) )  =  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) )
10 eqid 2443 . . . . 5  |-  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  { x } )  oF ( .r `  R
) g ) )  =  ( x  e.  K ,  g  e.  ( K  ^m  D
)  |->  ( ( D  X.  { x }
)  oF ( .r `  R ) g ) )
11 eqidd 2444 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) )  =  ( Xt_ `  ( D  X.  { ( TopOpen `  R ) } ) ) )
12 psrbas.i . . . . . 6  |-  ( ph  ->  I  e.  V )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  I  e.  V )
14 simpr 461 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  R  e. 
_V )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14psrval 17990 . . . 4  |-  ( (
ph  /\  R  e.  _V )  ->  S  =  ( { <. ( Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
1615fveq2d 5860 . . 3  |-  ( (
ph  /\  R  e.  _V )  ->  ( Base `  S )  =  (
Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
17 psrbas.b . . 3  |-  B  =  ( Base `  S
)
18 ovex 6309 . . . 4  |-  ( K  ^m  D )  e. 
_V
19 psrvalstr 17991 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) Struct  <. 1 ,  9 >.
20 baseid 14660 . . . . 5  |-  Base  = Slot  ( Base `  ndx )
21 snsstp1 4166 . . . . . 6  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. }  C_  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }
22 ssun1 3652 . . . . . 6  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  C_  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
2321, 22sstri 3498 . . . . 5  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. }  C_  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
2419, 20, 23strfv 14648 . . . 4  |-  ( ( K  ^m  D )  e.  _V  ->  ( K  ^m  D )  =  ( Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
2518, 24ax-mp 5 . . 3  |-  ( K  ^m  D )  =  ( Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
2616, 17, 253eqtr4g 2509 . 2  |-  ( (
ph  /\  R  e.  _V )  ->  B  =  ( K  ^m  D
) )
27 reldmpsr 17989 . . . . . . . 8  |-  Rel  dom mPwSer
2827ovprc2 6313 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( I mPwSer  R )  =  (/) )
2928adantl 466 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
I mPwSer  R )  =  (/) )
301, 29syl5eq 2496 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  S  =  (/) )
3130fveq2d 5860 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( Base `  S )  =  ( Base `  (/) ) )
32 base0 14653 . . . 4  |-  (/)  =  (
Base `  (/) )
3331, 17, 323eqtr4g 2509 . . 3  |-  ( (
ph  /\  -.  R  e.  _V )  ->  B  =  (/) )
34 fvprc 5850 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
3534adantl 466 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( Base `  R )  =  (/) )
362, 35syl5eq 2496 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  K  =  (/) )
37 0nn0 10817 . . . . . . . 8  |-  0  e.  NN0
3837a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  -.  R  e.  _V )  /\  x  e.  I
)  ->  0  e.  NN0 )
39 eqid 2443 . . . . . . 7  |-  ( x  e.  I  |->  0 )  =  ( x  e.  I  |->  0 )
4038, 39fmptd 6040 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
x  e.  I  |->  0 ) : I --> NN0 )
41 0fin 7749 . . . . . . 7  |-  (/)  e.  Fin
42 nn0suppOLD 10857 . . . . . . . . 9  |-  ( ( x  e.  I  |->  0 ) : I --> NN0  ->  ( `' ( x  e.  I  |->  0 ) "
( _V  \  {
0 } ) )  =  ( `' ( x  e.  I  |->  0 ) " NN ) )
4340, 42syl 16 . . . . . . . 8  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) "
( _V  \  {
0 } ) )  =  ( `' ( x  e.  I  |->  0 ) " NN ) )
44 eqidd 2444 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  R  e.  _V )  /\  x  e.  (
I  \  (/) ) )  ->  0  =  0 )
4544suppss2OLD 6515 . . . . . . . 8  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) "
( _V  \  {
0 } ) ) 
C_  (/) )
4643, 45eqsstr3d 3524 . . . . . . 7  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) " NN )  C_  (/) )
47 ssfi 7742 . . . . . . 7  |-  ( (
(/)  e.  Fin  /\  ( `' ( x  e.  I  |->  0 ) " NN )  C_  (/) )  -> 
( `' ( x  e.  I  |->  0 )
" NN )  e. 
Fin )
4841, 46, 47sylancr 663 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin )
496psrbag 17992 . . . . . . . 8  |-  ( I  e.  V  ->  (
( x  e.  I  |->  0 )  e.  D  <->  ( ( x  e.  I  |->  0 ) : I --> NN0  /\  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin )
) )
5012, 49syl 16 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  I  |->  0 )  e.  D  <->  ( ( x  e.  I  |->  0 ) : I --> NN0  /\  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin ) ) )
5150adantr 465 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
( x  e.  I  |->  0 )  e.  D  <->  ( ( x  e.  I  |->  0 ) : I --> NN0  /\  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin )
) )
5240, 48, 51mpbir2and 922 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
x  e.  I  |->  0 )  e.  D )
53 ne0i 3776 . . . . 5  |-  ( ( x  e.  I  |->  0 )  e.  D  ->  D  =/=  (/) )
5452, 53syl 16 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  D  =/=  (/) )
55 fvex 5866 . . . . . 6  |-  ( Base `  R )  e.  _V
562, 55eqeltri 2527 . . . . 5  |-  K  e. 
_V
57 ovex 6309 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
5857rabex 4588 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
596, 58eqeltri 2527 . . . . 5  |-  D  e. 
_V
6056, 59map0 7461 . . . 4  |-  ( ( K  ^m  D )  =  (/)  <->  ( K  =  (/)  /\  D  =/=  (/) ) )
6136, 54, 60sylanbrc 664 . . 3  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( K  ^m  D )  =  (/) )
6233, 61eqtr4d 2487 . 2  |-  ( (
ph  /\  -.  R  e.  _V )  ->  B  =  ( K  ^m  D ) )
6326, 62pm2.61dan 791 1  |-  ( ph  ->  B  =  ( K  ^m  D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   {crab 2797   _Vcvv 3095    \ cdif 3458    u. cun 3459    C_ wss 3461   (/)c0 3770   {csn 4014   {ctp 4018   <.cop 4020   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   `'ccnv 4988    |` cres 4991   "cima 4992   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283    oFcof 6523    oRcofr 6524    ^m cmap 7422   Fincfn 7518   0cc0 9495   1c1 9496    <_ cle 9632    - cmin 9810   NNcn 10543   9c9 10599   NN0cn0 10802   ndxcnx 14611   Basecbs 14614   +g cplusg 14679   .rcmulr 14680  Scalarcsca 14682   .scvsca 14683  TopSetcts 14685   TopOpenctopn 14801   Xt_cpt 14818    gsumg cgsu 14820   mPwSer cmps 17979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-plusg 14692  df-mulr 14693  df-sca 14695  df-vsca 14696  df-tset 14698  df-psr 17984
This theorem is referenced by: (None)
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