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Theorem pellfundglb 26640
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 26635 . . . . . . 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 4176 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  <  A )
5 pellfundre 26636 . . . . . . . 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 2463 . . . . . 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 9153 . . . . 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 3372 . . . . . 6  |-  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  (Pell14QR `  D )
12 pell14qrre 26612 . . . . . . . . 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 3298 . . . . . . 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 3303 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  RR )
17 pell1qrss14 26623 . . . . . . . 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 26634 . . . . . . . 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 3352 . . . . . . 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 3591 . . . . . 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 26633 . . . . . 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 2661 . . 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 4158 . . . . . . . 8  |-  ( a  =  x  ->  (
1  <  a  <->  1  <  x ) )
3231elrab 3036 . . . . . . 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 9024 . . . . . . . . . . 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 26612 . . . . . . . . . . 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 9154 . . . . . . . . 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 26627 . . . . . . . . 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 3290 . . . . . . 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 26639 . . . . . . 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 3293 . . . . . . . 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 9153 . . . . . . 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 2759 . 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 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   E.wrex 2651   {crab 2654    \ cdif 3261    C_ wss 3264   (/)c0 3572   class class class wbr 4154   `'ccnv 4818   ` cfv 5395   supcsup 7381   RRcr 8923   1c1 8925    < clt 9054    <_ cle 9055   NNcn 9933  ◻NNcsquarenn 26591  Pell1QRcpell1qr 26592  Pell14QRcpell14qr 26594  PellFundcpellfund 26595
This theorem is referenced by:  pellfundex  26641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-omul 6666  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-acn 7763  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-ico 10855  df-fz 10977  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-dvds 12781  df-gcd 12935  df-numer 13055  df-denom 13056  df-squarenn 26596  df-pell1qr 26597  df-pell14qr 26598  df-pell1234qr 26599  df-pellfund 26600
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