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Theorem pellfundglb 26939
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 26934 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
213ad2ant1 978 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) )
3 simp3 959 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  <  A
)
42, 3eqbrtrrd 4226 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  <  A )
5 pellfundre 26935 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
653ad2ant1 978 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  e.  RR )
72, 6eqeltrrd 2510 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  e.  RR )
8 simp2 958 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  A  e.  RR )
97, 8ltnled 9212 . . . . 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 202 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  -.  A  <_  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
11 ssrab2 3420 . . . . . 6  |-  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  (Pell14QR `  D )
12 pell14qrre 26911 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D ) )  -> 
a  e.  RR )
1312ex 424 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell14QR `  D )  ->  a  e.  RR ) )
1413ssrdv 3346 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  RR )
15143ad2ant1 978 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (Pell14QR `  D )  C_  RR )
1611, 15syl5ss 3351 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  RR )
17 pell1qrss14 26922 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
18173ad2ant1 978 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (Pell1QR `  D )  C_  (Pell14QR `  D ) )
19 pellqrex 26933 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  E. a  e.  (Pell1QR `  D ) 1  < 
a )
20193ad2ant1 978 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. a  e.  (Pell1QR `  D )
1  <  a )
21 ssrexv 3400 . . . . . . 7  |-  ( (Pell1QR `  D )  C_  (Pell14QR `  D )  ->  ( E. a  e.  (Pell1QR `  D ) 1  < 
a  ->  E. a  e.  (Pell14QR `  D )
1  <  a )
)
2218, 20, 21sylc 58 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. a  e.  (Pell14QR `  D )
1  <  a )
23 rabn0 3639 . . . . . 6  |-  ( { a  e.  (Pell14QR `  D
)  |  1  < 
a }  =/=  (/)  <->  E. a  e.  (Pell14QR `  D )
1  <  a )
2422, 23sylibr 204 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  =/=  (/) )
25 infmrgelbi 26932 . . . . . 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 424 . . . . 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 1184 . . . 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 170 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  -.  A. x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } A  <_  x )
29 rexnal 2708 . . 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 204 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. x  e.  { a  e.  (Pell14QR `  D )  |  1  <  a }  -.  A  <_  x )
31 breq2 4208 . . . . . . . 8  |-  ( a  =  x  ->  (
1  <  a  <->  1  <  x ) )
3231elrab 3084 . . . . . . 7  |-  ( x  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )
33 simprl 733 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  (Pell14QR `  D
) )
34 1re 9082 . . . . . . . . . . 11  |-  1  e.  RR
3534a1i 11 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  e.  RR )
36 simpl1 960 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  D  e.  ( NN  \NN )
)
37 pell14qrre 26911 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  x  e.  (Pell14QR `  D ) )  ->  x  e.  RR )
3836, 33, 37syl2anc 643 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  RR )
39 simprr 734 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  <  x )
4035, 38, 39ltled 9213 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  <_  x )
4133, 40jca 519 . . . . . . . 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 26926 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
( x  e.  (Pell1QR `  D )  <->  ( x  e.  (Pell14QR `  D )  /\  1  <_  x ) ) )
4336, 42syl 16 . . . . . . . 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 224 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  (Pell1QR `  D
) )
4532, 44sylan2b 462 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } )  ->  x  e.  (Pell1QR `  D
) )
4645adantrr 698 . . . . 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 960 . . . . . . 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 733 . . . . . . . 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 3338 . . . . . . 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 448 . . . . . . . . . . 11  |-  ( ( x  e.  (Pell14QR `  D
)  /\  1  <  x )  ->  1  <  x )
5150a1i 11 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (
( x  e.  (Pell14QR `  D )  /\  1  <  x )  ->  1  <  x ) )
5232, 51syl5bi 209 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (
x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  ->  1  <  x ) )
5352imp 419 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } )  -> 
1  <  x )
5453adantrr 698 . . . . . . 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 26938 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  x  e.  (Pell14QR `  D )  /\  1  <  x )  ->  (PellFund `  D )  <_  x
)
5647, 49, 54, 55syl3anc 1184 . . . . . 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 734 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  -.  A  <_  x )
5815adantr 452 . . . . . . . . 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 3341 . . . . . . . 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 961 . . . . . . . 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 9212 . . . . . . 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 224 . . . . . 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 519 . . . . 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 519 . . . 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 424 . . 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 2807 . 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 15 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    \ cdif 3309    C_ wss 3312   (/)c0 3620   class class class wbr 4204   `'ccnv 4869   ` cfv 5446   supcsup 7437   RRcr 8981   1c1 8983    < clt 9112    <_ cle 9113   NNcn 9992  ◻NNcsquarenn 26890  Pell1QRcpell1qr 26891  Pell14QRcpell14qr 26893  PellFundcpellfund 26894
This theorem is referenced by:  pellfundex  26940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-acn 7821  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-ico 10914  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-numer 13119  df-denom 13120  df-squarenn 26895  df-pell1qr 26896  df-pell14qr 26897  df-pell1234qr 26898  df-pellfund 26899
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