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Theorem pellfundglb 29367
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 29362 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
213ad2ant1 1009 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) )
3 simp3 990 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  <  A
)
42, 3eqbrtrrd 4415 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  <  A )
5 pellfundre 29363 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
653ad2ant1 1009 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  e.  RR )
72, 6eqeltrrd 2540 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  e.  RR )
8 simp2 989 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  A  e.  RR )
97, 8ltnled 9625 . . . . 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 210 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  -.  A  <_  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
11 ssrab2 3538 . . . . . 6  |-  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  (Pell14QR `  D )
12 pell14qrre 29339 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D ) )  -> 
a  e.  RR )
1312ex 434 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell14QR `  D )  ->  a  e.  RR ) )
1413ssrdv 3463 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  RR )
15143ad2ant1 1009 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (Pell14QR `  D )  C_  RR )
1611, 15syl5ss 3468 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  RR )
17 pell1qrss14 29350 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
18173ad2ant1 1009 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (Pell1QR `  D )  C_  (Pell14QR `  D ) )
19 pellqrex 29361 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  E. a  e.  (Pell1QR `  D ) 1  < 
a )
20193ad2ant1 1009 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. a  e.  (Pell1QR `  D )
1  <  a )
21 ssrexv 3518 . . . . . . 7  |-  ( (Pell1QR `  D )  C_  (Pell14QR `  D )  ->  ( E. a  e.  (Pell1QR `  D ) 1  < 
a  ->  E. a  e.  (Pell14QR `  D )
1  <  a )
)
2218, 20, 21sylc 60 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. a  e.  (Pell14QR `  D )
1  <  a )
23 rabn0 3758 . . . . . 6  |-  ( { a  e.  (Pell14QR `  D
)  |  1  < 
a }  =/=  (/)  <->  E. a  e.  (Pell14QR `  D )
1  <  a )
2422, 23sylibr 212 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  =/=  (/) )
25 infmrgelbi 29360 . . . . . 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 434 . . . . 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 1219 . . . 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 177 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  -.  A. x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } A  <_  x )
29 rexnal 2847 . . 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 212 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. x  e.  { a  e.  (Pell14QR `  D )  |  1  <  a }  -.  A  <_  x )
31 breq2 4397 . . . . . . . 8  |-  ( a  =  x  ->  (
1  <  a  <->  1  <  x ) )
3231elrab 3217 . . . . . . 7  |-  ( x  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )
33 simprl 755 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  (Pell14QR `  D
) )
34 1re 9489 . . . . . . . . . . 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 991 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  D  e.  ( NN  \NN )
)
37 pell14qrre 29339 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  x  e.  (Pell14QR `  D ) )  ->  x  e.  RR )
3836, 33, 37syl2anc 661 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  RR )
39 simprr 756 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  <  x )
4035, 38, 39ltled 9626 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  <_  x )
4133, 40jca 532 . . . . . . . 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 29354 . . . . . . . . 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 232 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  (Pell1QR `  D
) )
4532, 44sylan2b 475 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } )  ->  x  e.  (Pell1QR `  D
) )
4645adantrr 716 . . . . 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 991 . . . . . . 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 755 . . . . . . . 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 3455 . . . . . . 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 461 . . . . . . . . . . 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 217 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (
x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  ->  1  <  x ) )
5352imp 429 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } )  -> 
1  <  x )
5453adantrr 716 . . . . . . 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 29366 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  x  e.  (Pell14QR `  D )  /\  1  <  x )  ->  (PellFund `  D )  <_  x
)
5647, 49, 54, 55syl3anc 1219 . . . . . 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 756 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  -.  A  <_  x )
5815adantr 465 . . . . . . . . 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 3458 . . . . . . . 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 992 . . . . . . . 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 9625 . . . . . . 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 232 . . . . . 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 532 . . . . 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 532 . . . 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 434 . . 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 2924 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   {crab 2799    \ cdif 3426    C_ wss 3429   (/)c0 3738   class class class wbr 4393   `'ccnv 4940   ` cfv 5519   supcsup 7794   RRcr 9385   1c1 9387    < clt 9522    <_ cle 9523   NNcn 10426  ◻NNcsquarenn 29318  Pell1QRcpell1qr 29319  Pell14QRcpell14qr 29321  PellFundcpellfund 29322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-omul 7028  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-acn 8216  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-q 11058  df-rp 11096  df-ico 11410  df-fz 11548  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-dvds 13647  df-gcd 13802  df-numer 13924  df-denom 13925  df-squarenn 29323  df-pell1qr 29324  df-pell14qr 29325  df-pell1234qr 29326  df-pellfund 29327
This theorem is referenced by:  pellfundex  29368
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