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Theorem pellfundex 29152
Description: The fundamental solution as an infimum is itself a solution, showing that the solution set is discrete.

Since the fundamental solution is an infimum, there must be an element ge to Fund and lt 2*Fund. If this element is equal to the fundamental solution we're done, otherwise use the infimum again to find another element which must be ge Fund and lt the first element; their ratio is a group element in (1,2), contradicting pell14qrgapw 29142. (Contributed by Stefan O'Rear, 18-Sep-2014.)

Assertion
Ref Expression
pellfundex  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
)

Proof of Theorem pellfundex
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2re 10387 . . . 4  |-  2  e.  RR
2 pellfundre 29147 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
3 remulcl 9363 . . . 4  |-  ( ( 2  e.  RR  /\  (PellFund `  D )  e.  RR )  ->  (
2  x.  (PellFund `  D
) )  e.  RR )
41, 2, 3sylancr 658 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( 2  x.  (PellFund `  D ) )  e.  RR )
5 0re 9382 . . . . . . . 8  |-  0  e.  RR
65a1i 11 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
0  e.  RR )
7 1re 9381 . . . . . . . 8  |-  1  e.  RR
87a1i 11 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  RR )
9 0lt1 9858 . . . . . . . 8  |-  0  <  1
109a1i 11 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
0  <  1 )
11 pellfundgt1 29149 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
1  <  (PellFund `  D
) )
126, 8, 2, 10, 11lttrd 9528 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
0  <  (PellFund `  D
) )
132, 12elrpd 11021 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR+ )
142, 13ltaddrpd 11052 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  < 
( (PellFund `  D )  +  (PellFund `  D )
) )
152recnd 9408 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  CC )
16152timesd 10563 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( 2  x.  (PellFund `  D ) )  =  ( (PellFund `  D
)  +  (PellFund `  D
) ) )
1714, 16breqtrrd 4315 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  < 
( 2  x.  (PellFund `  D ) ) )
18 pellfundglb 29151 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  ( 2  x.  (PellFund `  D )
)  e.  RR  /\  (PellFund `  D )  < 
( 2  x.  (PellFund `  D ) ) )  ->  E. a  e.  (Pell1QR `  D ) ( (PellFund `  D )  <_  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )
194, 17, 18mpd3an23 1311 . 2  |-  ( D  e.  ( NN  \NN )  ->  E. a  e.  (Pell1QR `  D ) ( (PellFund `  D )  <_  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )
202adantr 462 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
(PellFund `  D )  e.  RR )
21 pell1qrss14 29134 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
2221sselda 3353 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
a  e.  (Pell14QR `  D
) )
23 pell14qrre 29123 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D ) )  -> 
a  e.  RR )
2422, 23syldan 467 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
a  e.  RR )
2520, 24leloed 9513 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <_  a  <->  ( (PellFund `  D
)  <  a  \/  (PellFund `  D )  =  a ) ) )
26 simpll 748 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  D  e.  ( NN  \NN ) )
2724adantr 462 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  a  e.  RR )
28 simprl 750 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  (PellFund `  D )  <  a )
29 pellfundglb 29151 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  RR  /\  (PellFund `  D )  <  a )  ->  E. b  e.  (Pell1QR `  D )
( (PellFund `  D )  <_  b  /\  b  < 
a ) )
3026, 27, 28, 29syl3anc 1213 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  E. b  e.  (Pell1QR `  D ) ( (PellFund `  D )  <_  b  /\  b  <  a ) )
31 simp-4l 760 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  ->  D  e.  ( NN  \NN )
)
32 simp-4r 761 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
a  e.  (Pell1QR `  D
) )
33 simplr 749 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
b  e.  (Pell1QR `  D
) )
34 simprr 751 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
b  <  a )
3524ad3antrrr 724 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
a  e.  RR )
364adantr 462 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( 2  x.  (PellFund `  D ) )  e.  RR )
3736ad3antrrr 724 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
( 2  x.  (PellFund `  D ) )  e.  RR )
3821adantr 462 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
3938ad3antrrr 724 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
4039, 33sseldd 3354 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
b  e.  (Pell14QR `  D
) )
41 pell14qrre 29123 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  b  e.  (Pell14QR `  D ) )  -> 
b  e.  RR )
4231, 40, 41syl2anc 656 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
b  e.  RR )
43 remulcl 9363 . . . . . . . . . . . . 13  |-  ( ( 2  e.  RR  /\  b  e.  RR )  ->  ( 2  x.  b
)  e.  RR )
441, 42, 43sylancr 658 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
( 2  x.  b
)  e.  RR )
45 simprr 751 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  a  <  (
2  x.  (PellFund `  D
) ) )
4645ad2antrr 720 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
a  <  ( 2  x.  (PellFund `  D
) ) )
47 simprl 750 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
(PellFund `  D )  <_ 
b )
4820ad3antrrr 724 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
(PellFund `  D )  e.  RR )
491a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
2  e.  RR )
50 2pos 10409 . . . . . . . . . . . . . . 15  |-  0  <  2
5150a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
0  <  2 )
52 lemul2 10178 . . . . . . . . . . . . . 14  |-  ( ( (PellFund `  D )  e.  RR  /\  b  e.  RR  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
( (PellFund `  D )  <_  b  <->  ( 2  x.  (PellFund `  D )
)  <_  ( 2  x.  b ) ) )
5348, 42, 49, 51, 52syl112anc 1217 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
( (PellFund `  D )  <_  b  <->  ( 2  x.  (PellFund `  D )
)  <_  ( 2  x.  b ) ) )
5447, 53mpbid 210 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
( 2  x.  (PellFund `  D ) )  <_ 
( 2  x.  b
) )
5535, 37, 44, 46, 54ltletrd 9527 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
a  <  ( 2  x.  b ) )
56 simp1 983 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  D  e.  ( NN  \NN ) )
57213ad2ant1 1004 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  (Pell1QR `  D
)  C_  (Pell14QR `  D
) )
58 simp2l 1009 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  e.  (Pell1QR `  D ) )
5957, 58sseldd 3354 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  e.  (Pell14QR `  D ) )
60 simp2r 1010 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  (Pell1QR `  D ) )
6157, 60sseldd 3354 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  (Pell14QR `  D ) )
62 pell14qrdivcl 29131 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D )  /\  b  e.  (Pell14QR `  D )
)  ->  ( a  /  b )  e.  (Pell14QR `  D )
)
6356, 59, 61, 62syl3anc 1213 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( a  /  b )  e.  (Pell14QR `  D )
)
6456, 61, 41syl2anc 656 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  RR )
6564recnd 9408 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  CC )
6665mulid2d 9400 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( 1  x.  b )  =  b )
67 simp3l 1011 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  <  a )
6866, 67eqbrtrd 4309 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( 1  x.  b )  < 
a )
697a1i 11 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  1  e.  RR )
7056, 59, 23syl2anc 656 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  e.  RR )
71 pell14qrgt0 29125 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( NN 
\NN )  /\  b  e.  (Pell14QR `  D ) )  -> 
0  <  b )
7256, 61, 71syl2anc 656 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  0  <  b )
73 ltmuldiv 10198 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  RR  /\  a  e.  RR  /\  (
b  e.  RR  /\  0  <  b ) )  ->  ( ( 1  x.  b )  < 
a  <->  1  <  (
a  /  b ) ) )
7469, 70, 64, 72, 73syl112anc 1217 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( (
1  x.  b )  <  a  <->  1  <  ( a  /  b ) ) )
7568, 74mpbid 210 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  1  <  ( a  /  b ) )
76 simp3r 1012 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  <  ( 2  x.  b ) )
771a1i 11 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  2  e.  RR )
78 ltdivmul2 10203 . . . . . . . . . . . . . 14  |-  ( ( a  e.  RR  /\  2  e.  RR  /\  (
b  e.  RR  /\  0  <  b ) )  ->  ( ( a  /  b )  <  2  <->  a  <  (
2  x.  b ) ) )
7970, 77, 64, 72, 78syl112anc 1217 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( (
a  /  b )  <  2  <->  a  <  ( 2  x.  b ) ) )
8076, 79mpbird 232 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( a  /  b )  <  2 )
81 simprr 751 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  <  2
)
82 simpll 748 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  D  e.  ( NN  \NN ) )
83 simplr 749 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  e.  (Pell14QR `  D ) )
84 simprl 750 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  1  <  (
a  /  b ) )
85 pell14qrgapw 29142 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  / 
b )  e.  (Pell14QR `  D )  /\  1  <  ( a  /  b
) )  ->  2  <  ( a  /  b
) )
8682, 83, 84, 85syl3anc 1213 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  2  <  (
a  /  b ) )
87 pell14qrre 29123 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  / 
b )  e.  (Pell14QR `  D ) )  -> 
( a  /  b
)  e.  RR )
8887adantr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  e.  RR )
89 ltnsym 9469 . . . . . . . . . . . . . . 15  |-  ( ( 2  e.  RR  /\  ( a  /  b
)  e.  RR )  ->  ( 2  < 
( a  /  b
)  ->  -.  (
a  /  b )  <  2 ) )
901, 88, 89sylancr 658 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( 2  < 
( a  /  b
)  ->  -.  (
a  /  b )  <  2 ) )
9186, 90mpd 15 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  -.  ( a  /  b )  <  2 )
9281, 91pm2.21dd 174 . . . . . . . . . . . 12  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
)
9356, 63, 75, 80, 92syl22anc 1214 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  (PellFund `  D
)  e.  (Pell1QR `  D
) )
9431, 32, 33, 34, 55, 93syl122anc 1222 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D
) )  /\  (
(PellFund `  D )  < 
a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  /\  ( (PellFund `  D )  <_  b  /\  b  < 
a ) )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
)
9594ex 434 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  /\  b  e.  (Pell1QR `  D ) )  -> 
( ( (PellFund `  D
)  <_  b  /\  b  <  a )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
) )
9695rexlimdva 2839 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  ( E. b  e.  (Pell1QR `  D )
( (PellFund `  D )  <_  b  /\  b  < 
a )  ->  (PellFund `  D )  e.  (Pell1QR `  D ) ) )
9730, 96mpd 15 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
)
9897exp32 602 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <  a  ->  ( a  <  ( 2  x.  (PellFund `  D ) )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
) ) )
99 simp2 984 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  =  a )
100 simp1r 1008 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  a  e.  (Pell1QR `  D )
)
10199, 100eqeltrd 2515 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  e.  (Pell1QR `  D ) )
1021013exp 1181 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  =  a  ->  ( a  <  ( 2  x.  (PellFund `  D )
)  ->  (PellFund `  D
)  e.  (Pell1QR `  D
) ) ) )
10398, 102jaod 380 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( ( (PellFund `  D
)  <  a  \/  (PellFund `  D )  =  a )  ->  (
a  <  ( 2  x.  (PellFund `  D
) )  ->  (PellFund `  D )  e.  (Pell1QR `  D ) ) ) )
10425, 103sylbid 215 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <_  a  ->  ( a  <  ( 2  x.  (PellFund `  D ) )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
) ) )
105104imp3a 431 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( ( (PellFund `  D
)  <_  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  e.  (Pell1QR `  D ) ) )
106105rexlimdva 2839 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( E. a  e.  (Pell1QR `  D )
( (PellFund `  D )  <_  a  /\  a  < 
( 2  x.  (PellFund `  D ) ) )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
) )
10719, 106mpd 15 1  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   E.wrex 2714    \ cdif 3322    C_ wss 3325   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415    / cdiv 9989   NNcn 10318   2c2 10367  ◻NNcsquarenn 29102  Pell1QRcpell1qr 29103  Pell14QRcpell14qr 29105  PellFundcpellfund 29106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-omul 6921  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-ico 11302  df-fz 11434  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-numer 13809  df-denom 13810  df-squarenn 29107  df-pell1qr 29108  df-pell14qr 29109  df-pell1234qr 29110  df-pellfund 29111
This theorem is referenced by:  pellfund14  29164  pellfund14b  29165
  Copyright terms: Public domain W3C validator