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Theorem pellfundex 26963
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 26953. (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 10074 . . . 4  |-  2  e.  RR
2 pellfundre 26958 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
3 remulcl 9080 . . . 4  |-  ( ( 2  e.  RR  /\  (PellFund `  D )  e.  RR )  ->  (
2  x.  (PellFund `  D
) )  e.  RR )
41, 2, 3sylancr 646 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( 2  x.  (PellFund `  D ) )  e.  RR )
5 0re 9096 . . . . . . . 8  |-  0  e.  RR
65a1i 11 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
0  e.  RR )
7 1re 9095 . . . . . . . 8  |-  1  e.  RR
87a1i 11 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  RR )
9 0lt1 9555 . . . . . . . 8  |-  0  <  1
109a1i 11 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
0  <  1 )
11 pellfundgt1 26960 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
1  <  (PellFund `  D
) )
126, 8, 2, 10, 11lttrd 9236 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
0  <  (PellFund `  D
) )
132, 12elrpd 10651 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR+ )
142, 13ltaddrpd 10682 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  < 
( (PellFund `  D )  +  (PellFund `  D )
) )
152recnd 9119 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  CC )
16152timesd 10215 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( 2  x.  (PellFund `  D ) )  =  ( (PellFund `  D
)  +  (PellFund `  D
) ) )
1714, 16breqtrrd 4241 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  < 
( 2  x.  (PellFund `  D ) ) )
18 pellfundglb 26962 . . 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 1282 . 2  |-  ( D  e.  ( NN  \NN )  ->  E. a  e.  (Pell1QR `  D ) ( (PellFund `  D )  <_  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )
202adantr 453 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
(PellFund `  D )  e.  RR )
21 pell1qrss14 26945 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
2221sselda 3350 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
a  e.  (Pell14QR `  D
) )
23 pell14qrre 26934 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D ) )  -> 
a  e.  RR )
2422, 23syldan 458 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
a  e.  RR )
2520, 24leloed 9221 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <_  a  <->  ( (PellFund `  D
)  <  a  \/  (PellFund `  D )  =  a ) ) )
26 simpll 732 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  D  e.  ( NN  \NN ) )
2724adantr 453 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  a  e.  RR )
28 simprl 734 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  (PellFund `  D )  <  a )
29 pellfundglb 26962 . . . . . . . . 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 1185 . . . . . . . 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 744 . . . . . . . . . . 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 745 . . . . . . . . . . 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 733 . . . . . . . . . . 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 735 . . . . . . . . . . 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 712 . . . . . . . . . . . 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 453 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( 2  x.  (PellFund `  D ) )  e.  RR )
3736ad3antrrr 712 . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
3938ad3antrrr 712 . . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . . 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 26934 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  b  e.  (Pell14QR `  D ) )  -> 
b  e.  RR )
4231, 40, 41syl2anc 644 . . . . . . . . . . . . 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 9080 . . . . . . . . . . . . 13  |-  ( ( 2  e.  RR  /\  b  e.  RR )  ->  ( 2  x.  b
)  e.  RR )
441, 42, 43sylancr 646 . . . . . . . . . . . 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 735 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  a  <  (
2  x.  (PellFund `  D
) ) )
4645ad2antrr 708 . . . . . . . . . . . 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 734 . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . 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 10087 . . . . . . . . . . . . . . 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 9868 . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . 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 203 . . . . . . . . . . . 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 9235 . . . . . . . . . . 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 958 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  D  e.  ( NN  \NN ) )
57213ad2ant1 979 . . . . . . . . . . . . . 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 984 . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . 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 985 . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . 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 26942 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D )  /\  b  e.  (Pell14QR `  D )
)  ->  ( a  /  b )  e.  (Pell14QR `  D )
)
6356, 59, 61, 62syl3anc 1185 . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  RR )
6564recnd 9119 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  CC )
6665mulid2d 9111 . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  <  a )
6866, 67eqbrtrd 4235 . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  e.  RR )
71 pell14qrgt0 26936 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( NN 
\NN )  /\  b  e.  (Pell14QR `  D ) )  -> 
0  <  b )
7256, 61, 71syl2anc 644 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  0  <  b )
73 ltmuldiv 9885 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  RR  /\  a  e.  RR  /\  (
b  e.  RR  /\  0  <  b ) )  ->  ( ( 1  x.  b )  < 
a  <->  1  <  (
a  /  b ) ) )
7469, 70, 64, 72, 73syl112anc 1189 . . . . . . . . . . . . 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 203 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  1  <  ( a  /  b ) )
76 simp3r 987 . . . . . . . . . . . . 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 9890 . . . . . . . . . . . . . 14  |-  ( ( a  e.  RR  /\  2  e.  RR  /\  (
b  e.  RR  /\  0  <  b ) )  ->  ( ( a  /  b )  <  2  <->  a  <  (
2  x.  b ) ) )
7970, 77, 64, 72, 78syl112anc 1189 . . . . . . . . . . . . 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 225 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( a  /  b )  <  2 )
81 simprr 735 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  <  2
)
82 simpll 732 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  D  e.  ( NN  \NN ) )
83 simplr 733 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  e.  (Pell14QR `  D ) )
84 simprl 734 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  1  <  (
a  /  b ) )
85 pell14qrgapw 26953 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  / 
b )  e.  (Pell14QR `  D )  /\  1  <  ( a  /  b
) )  ->  2  <  ( a  /  b
) )
8682, 83, 84, 85syl3anc 1185 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  2  <  (
a  /  b ) )
87 pell14qrre 26934 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  / 
b )  e.  (Pell14QR `  D ) )  -> 
( a  /  b
)  e.  RR )
8887adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  e.  RR )
89 ltnsym 9177 . . . . . . . . . . . . . . 15  |-  ( ( 2  e.  RR  /\  ( a  /  b
)  e.  RR )  ->  ( 2  < 
( a  /  b
)  ->  -.  (
a  /  b )  <  2 ) )
901, 88, 89sylancr 646 . . . . . . . . . . . . . 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 102 . . . . . . . . . . . 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 1186 . . . . . . . . . . 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 1194 . . . . . . . . . 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 425 . . . . . . . . 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 2832 . . . . . . . 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 590 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <  a  ->  ( a  <  ( 2  x.  (PellFund `  D ) )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
) ) )
99 simp2 959 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  =  a )
100 simp1r 983 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  a  e.  (Pell1QR `  D )
)
10199, 100eqeltrd 2512 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  e.  (Pell1QR `  D ) )
1021013exp 1153 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  =  a  ->  ( a  <  ( 2  x.  (PellFund `  D )
)  ->  (PellFund `  D
)  e.  (Pell1QR `  D
) ) ) )
10398, 102jaod 371 . . . . 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 208 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <_  a  ->  ( a  <  ( 2  x.  (PellFund `  D ) )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
) ) )
105104imp3a 422 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( ( (PellFund `  D
)  <_  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  e.  (Pell1QR `  D ) ) )
106105rexlimdva 2832 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708    \ cdif 3319    C_ wss 3322   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000    < clt 9125    <_ cle 9126    / cdiv 9682   NNcn 10005   2c2 10054  ◻NNcsquarenn 26913  Pell1QRcpell1qr 26914  Pell14QRcpell14qr 26916  PellFundcpellfund 26917
This theorem is referenced by:  pellfund14  26975  pellfund14b  26976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-omul 6732  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-acn 7834  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-ico 10927  df-fz 11049  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-dvds 12858  df-gcd 13012  df-numer 13132  df-denom 13133  df-squarenn 26918  df-pell1qr 26919  df-pell14qr 26920  df-pell1234qr 26921  df-pellfund 26922
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