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Theorem pellfundglb 27073
Description: If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pellfundglb  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. x  e.  (Pell1QR `  D )
( (PellFund `  D )  <_  x  /\  x  < 
A ) )
Distinct variable groups:    x, D    x, A

Proof of Theorem pellfundglb
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 pellfundval 27068 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
213ad2ant1 976 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) )
3 simp3 957 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  <  A
)
42, 3eqbrtrrd 4061 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  <  A )
5 pellfundre 27069 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
653ad2ant1 976 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  e.  RR )
72, 6eqeltrrd 2371 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  e.  RR )
8 simp2 956 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  A  e.  RR )
97, 8ltnled 8982 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  ( sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  <  A  <->  -.  A  <_  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) ) )
104, 9mpbid 201 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  -.  A  <_  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
11 ssrab2 3271 . . . . . 6  |-  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  (Pell14QR `  D )
12 pell14qrre 27045 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D ) )  -> 
a  e.  RR )
1312ex 423 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell14QR `  D )  ->  a  e.  RR ) )
1413ssrdv 3198 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  RR )
15143ad2ant1 976 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (Pell14QR `  D )  C_  RR )
1611, 15syl5ss 3203 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  RR )
17 pell1qrss14 27056 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
18173ad2ant1 976 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (Pell1QR `  D )  C_  (Pell14QR `  D ) )
19 pellqrex 27067 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  E. a  e.  (Pell1QR `  D ) 1  < 
a )
20193ad2ant1 976 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. a  e.  (Pell1QR `  D )
1  <  a )
21 ssrexv 3251 . . . . . . 7  |-  ( (Pell1QR `  D )  C_  (Pell14QR `  D )  ->  ( E. a  e.  (Pell1QR `  D ) 1  < 
a  ->  E. a  e.  (Pell14QR `  D )
1  <  a )
)
2218, 20, 21sylc 56 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. a  e.  (Pell14QR `  D )
1  <  a )
23 rabn0 3487 . . . . . 6  |-  ( { a  e.  (Pell14QR `  D
)  |  1  < 
a }  =/=  (/)  <->  E. a  e.  (Pell14QR `  D )
1  <  a )
2422, 23sylibr 203 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  =/=  (/) )
25 infmrgelbi 27066 . . . . . 6  |-  ( ( ( { a  e.  (Pell14QR `  D )  |  1  <  a }  C_  RR  /\  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  =/=  (/)  /\  A  e.  RR )  /\  A. x  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } A  <_  x )  ->  A  <_  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) )
2625ex 423 . . . . 5  |-  ( ( { a  e.  (Pell14QR `  D )  |  1  <  a }  C_  RR  /\  { a  e.  (Pell14QR `  D )  |  1  <  a }  =/=  (/)  /\  A  e.  RR )  ->  ( A. x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } A  <_  x  ->  A  <_  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) ) )
2716, 24, 8, 26syl3anc 1182 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  ( A. x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } A  <_  x  ->  A  <_  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) ) )
2810, 27mtod 168 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  -.  A. x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } A  <_  x )
29 rexnal 2567 . . 3  |-  ( E. x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  -.  A  <_  x  <->  -.  A. x  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } A  <_  x )
3028, 29sylibr 203 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. x  e.  { a  e.  (Pell14QR `  D )  |  1  <  a }  -.  A  <_  x )
31 breq2 4043 . . . . . . . 8  |-  ( a  =  x  ->  (
1  <  a  <->  1  <  x ) )
3231elrab 2936 . . . . . . 7  |-  ( x  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )
33 simprl 732 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  (Pell14QR `  D
) )
34 1re 8853 . . . . . . . . . . 11  |-  1  e.  RR
3534a1i 10 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  e.  RR )
36 simpl1 958 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  D  e.  ( NN  \NN )
)
37 pell14qrre 27045 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  x  e.  (Pell14QR `  D ) )  ->  x  e.  RR )
3836, 33, 37syl2anc 642 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  RR )
39 simprr 733 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  <  x )
4035, 38, 39ltled 8983 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  <_  x )
4133, 40jca 518 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
( x  e.  (Pell14QR `  D )  /\  1  <_  x ) )
42 elpell1qr2 27060 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
( x  e.  (Pell1QR `  D )  <->  ( x  e.  (Pell14QR `  D )  /\  1  <_  x ) ) )
4336, 42syl 15 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
( x  e.  (Pell1QR `  D )  <->  ( x  e.  (Pell14QR `  D )  /\  1  <_  x ) ) )
4441, 43mpbird 223 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  (Pell1QR `  D
) )
4532, 44sylan2b 461 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } )  ->  x  e.  (Pell1QR `  D
) )
4645adantrr 697 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  x  e.  (Pell1QR `  D ) )
47 simpl1 958 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  D  e.  ( NN  \NN ) )
48 simprl 732 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a } )
4911, 48sseldi 3191 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  x  e.  (Pell14QR `  D ) )
50 simpr 447 . . . . . . . . . . 11  |-  ( ( x  e.  (Pell14QR `  D
)  /\  1  <  x )  ->  1  <  x )
5150a1i 10 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (
( x  e.  (Pell14QR `  D )  /\  1  <  x )  ->  1  <  x ) )
5232, 51syl5bi 208 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (
x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  ->  1  <  x ) )
5352imp 418 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } )  -> 
1  <  x )
5453adantrr 697 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  1  <  x
)
55 pellfundlb 27072 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  x  e.  (Pell14QR `  D )  /\  1  <  x )  ->  (PellFund `  D )  <_  x
)
5647, 49, 54, 55syl3anc 1182 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  (PellFund `  D )  <_  x )
57 simprr 733 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  -.  A  <_  x )
5815adantr 451 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  (Pell14QR `  D )  C_  RR )
5958, 49sseldd 3194 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  x  e.  RR )
60 simpl2 959 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  A  e.  RR )
6159, 60ltnled 8982 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  ( x  < 
A  <->  -.  A  <_  x ) )
6257, 61mpbird 223 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  x  <  A
)
6356, 62jca 518 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  ( (PellFund `  D
)  <_  x  /\  x  <  A ) )
6446, 63jca 518 . . . 4  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  ( x  e.  (Pell1QR `  D )  /\  ( (PellFund `  D
)  <_  x  /\  x  <  A ) ) )
6564ex 423 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (
( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x )  -> 
( x  e.  (Pell1QR `  D )  /\  (
(PellFund `  D )  <_  x  /\  x  <  A
) ) ) )
6665reximdv2 2665 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  ( E. x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  -.  A  <_  x  ->  E. x  e.  (Pell1QR `  D )
( (PellFund `  D )  <_  x  /\  x  < 
A ) ) )
6730, 66mpd 14 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. x  e.  (Pell1QR `  D )
( (PellFund `  D )  <_  x  /\  x  < 
A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    C_ wss 3165   (/)c0 3468   class class class wbr 4039   `'ccnv 4704   ` cfv 5271   supcsup 7209   RRcr 8752   1c1 8754    < clt 8883    <_ cle 8884   NNcn 9762  ◻NNcsquarenn 27024  Pell1QRcpell1qr 27025  Pell14QRcpell14qr 27027  PellFundcpellfund 27028
This theorem is referenced by:  pellfundex  27074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-ico 10678  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-numer 12822  df-denom 12823  df-squarenn 27029  df-pell1qr 27030  df-pell14qr 27031  df-pell1234qr 27032  df-pellfund 27033
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