Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pellfundglb Structured version   Unicode version

Theorem pellfundglb 31060
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 31055 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
213ad2ant1 1015 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) )
3 simp3 996 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  <  A
)
42, 3eqbrtrrd 4461 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  <  A )
5 pellfundre 31056 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
653ad2ant1 1015 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (PellFund `  D )  e.  RR )
72, 6eqeltrrd 2543 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  e.  RR )
8 simp2 995 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  A  e.  RR )
97, 8ltnled 9721 . . . . 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 3571 . . . . . 6  |-  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  (Pell14QR `  D )
12 pell14qrre 31032 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D ) )  -> 
a  e.  RR )
1312ex 432 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell14QR `  D )  ->  a  e.  RR ) )
1413ssrdv 3495 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  RR )
15143ad2ant1 1015 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (Pell14QR `  D )  C_  RR )
1611, 15syl5ss 3500 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  RR )
17 pell1qrss14 31043 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
18173ad2ant1 1015 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (Pell1QR `  D )  C_  (Pell14QR `  D ) )
19 pellqrex 31054 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  E. a  e.  (Pell1QR `  D ) 1  < 
a )
20193ad2ant1 1015 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. a  e.  (Pell1QR `  D )
1  <  a )
21 ssrexv 3551 . . . . . . 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 3804 . . . . . 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 31053 . . . . . 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 432 . . . . 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 1226 . . . 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 2902 . . 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 4443 . . . . . . . 8  |-  ( a  =  x  ->  (
1  <  a  <->  1  <  x ) )
3231elrab 3254 . . . . . . 7  |-  ( x  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )
33 simprl 754 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  (Pell14QR `  D
) )
34 1red 9600 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  e.  RR )
35 simpl1 997 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  D  e.  ( NN  \NN )
)
36 pell14qrre 31032 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  x  e.  (Pell14QR `  D ) )  ->  x  e.  RR )
3735, 33, 36syl2anc 659 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  RR )
38 simprr 755 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  <  x )
3934, 37, 38ltled 9722 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
1  <_  x )
4033, 39jca 530 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  -> 
( x  e.  (Pell14QR `  D )  /\  1  <_  x ) )
41 elpell1qr2 31047 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
( x  e.  (Pell1QR `  D )  <->  ( x  e.  (Pell14QR `  D )  /\  1  <_  x ) ) )
4235, 41syl 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 ) ) )
4340, 42mpbird 232 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  (Pell14QR `  D )  /\  1  <  x ) )  ->  x  e.  (Pell1QR `  D
) )
4432, 43sylan2b 473 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } )  ->  x  e.  (Pell1QR `  D
) )
4544adantrr 714 . . . . 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 ) )
46 simpl1 997 . . . . . . 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 ) )
47 simprl 754 . . . . . . . 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 } )
4811, 47sseldi 3487 . . . . . . 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 ) )
49 simpr 459 . . . . . . . . . . 11  |-  ( ( x  e.  (Pell14QR `  D
)  /\  1  <  x )  ->  1  <  x )
5049a1i 11 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (
( x  e.  (Pell14QR `  D )  /\  1  <  x )  ->  1  <  x ) )
5132, 50syl5bi 217 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  (
x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  ->  1  <  x ) )
5251imp 427 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  x  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } )  -> 
1  <  x )
5352adantrr 714 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  1  <  x
)
54 pellfundlb 31059 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  x  e.  (Pell14QR `  D )  /\  1  <  x )  ->  (PellFund `  D )  <_  x
)
5546, 48, 53, 54syl3anc 1226 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  (PellFund `  D )  <_  x )
56 simprr 755 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  -.  A  <_  x )
5715adantr 463 . . . . . . . . 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 )
5857, 48sseldd 3490 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  x  e.  RR )
59 simpl2 998 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  A  e.  RR )
6058, 59ltnled 9721 . . . . . . 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 ) )
6156, 60mpbird 232 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D
)  <  A )  /\  ( x  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  /\  -.  A  <_  x ) )  ->  x  <  A
)
6255, 61jca 530 . . . . 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 ) )
6345, 62jca 530 . . . 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 ) ) )
6463ex 432 . . 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
) ) ) )
6564reximdv2 2925 . 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 ) ) )
6630, 65mpd 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808    \ cdif 3458    C_ wss 3461   (/)c0 3783   class class class wbr 4439   `'ccnv 4987   ` cfv 5570   supcsup 7892   RRcr 9480   1c1 9482    < clt 9617    <_ cle 9618   NNcn 10531  ◻NNcsquarenn 31011  Pell1QRcpell1qr 31012  Pell14QRcpell14qr 31014  PellFundcpellfund 31015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-acn 8314  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-ico 11538  df-fz 11676  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-dvds 14071  df-gcd 14229  df-numer 14352  df-denom 14353  df-squarenn 31016  df-pell1qr 31017  df-pell14qr 31018  df-pell1234qr 31019  df-pellfund 31020
This theorem is referenced by:  pellfundex  31061
  Copyright terms: Public domain W3C validator