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Theorem pellfundex 26294
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 26284. (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 9905 . . . 4  |-  2  e.  RR
2 pellfundre 26289 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
3 remulcl 8912 . . . 4  |-  ( ( 2  e.  RR  /\  (PellFund `  D )  e.  RR )  ->  (
2  x.  (PellFund `  D
) )  e.  RR )
41, 2, 3sylancr 644 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( 2  x.  (PellFund `  D ) )  e.  RR )
5 0re 8928 . . . . . . . 8  |-  0  e.  RR
65a1i 10 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
0  e.  RR )
7 1re 8927 . . . . . . . 8  |-  1  e.  RR
87a1i 10 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  RR )
9 0lt1 9386 . . . . . . . 8  |-  0  <  1
109a1i 10 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
0  <  1 )
11 pellfundgt1 26291 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
1  <  (PellFund `  D
) )
126, 8, 2, 10, 11lttrd 9067 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
0  <  (PellFund `  D
) )
132, 12elrpd 10480 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR+ )
142, 13ltaddrpd 10511 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  < 
( (PellFund `  D )  +  (PellFund `  D )
) )
152recnd 8951 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  CC )
16152timesd 10046 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( 2  x.  (PellFund `  D ) )  =  ( (PellFund `  D
)  +  (PellFund `  D
) ) )
1714, 16breqtrrd 4130 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  < 
( 2  x.  (PellFund `  D ) ) )
18 pellfundglb 26293 . . 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 1279 . 2  |-  ( D  e.  ( NN  \NN )  ->  E. a  e.  (Pell1QR `  D ) ( (PellFund `  D )  <_  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )
202adantr 451 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
(PellFund `  D )  e.  RR )
21 pell1qrss14 26276 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
2221sselda 3256 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
a  e.  (Pell14QR `  D
) )
23 pell14qrre 26265 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D ) )  -> 
a  e.  RR )
2422, 23syldan 456 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
a  e.  RR )
2520, 24leloed 9052 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <_  a  <->  ( (PellFund `  D
)  <  a  \/  (PellFund `  D )  =  a ) ) )
26 simpll 730 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  D  e.  ( NN  \NN ) )
2724adantr 451 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  a  e.  RR )
28 simprl 732 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  (PellFund `  D )  <  a )
29 pellfundglb 26293 . . . . . . . . 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 1182 . . . . . . . 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 simpl 443 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  ->  D  e.  ( NN  \NN )
)
3231ad3antrrr 710 . . . . . . . . . . 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 )
)
33 simpr 447 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
a  e.  (Pell1QR `  D
) )
3433ad3antrrr 710 . . . . . . . . . . 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
) )
35 simplr 731 . . . . . . . . . . 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
) )
36 simprr 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  <  a )
3724ad3antrrr 710 . . . . . . . . . . . 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 )
384adantr 451 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( 2  x.  (PellFund `  D ) )  e.  RR )
3938ad3antrrr 710 . . . . . . . . . . . 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 )
4021adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
4140ad3antrrr 710 . . . . . . . . . . . . . . 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 ) )
4241, 35sseldd 3257 . . . . . . . . . . . . . 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
) )
43 pell14qrre 26265 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  b  e.  (Pell14QR `  D ) )  -> 
b  e.  RR )
4432, 42, 43syl2anc 642 . . . . . . . . . . . . 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 )
45 remulcl 8912 . . . . . . . . . . . . 13  |-  ( ( 2  e.  RR  /\  b  e.  RR )  ->  ( 2  x.  b
)  e.  RR )
461, 44, 45sylancr 644 . . . . . . . . . . . 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 )
47 simprr 733 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  a  <  (
2  x.  (PellFund `  D
) ) )
4847ad2antrr 706 . . . . . . . . . . . 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
) ) )
49 simprl 732 . . . . . . . . . . . . 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 )
5020ad3antrrr 710 . . . . . . . . . . . . . 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 )
511a1i 10 . . . . . . . . . . . . . 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 )
52 2pos 9918 . . . . . . . . . . . . . . 15  |-  0  <  2
5352a1i 10 . . . . . . . . . . . . . 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 )
54 lemul2 9699 . . . . . . . . . . . . . 14  |-  ( ( (PellFund `  D )  e.  RR  /\  b  e.  RR  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
( (PellFund `  D )  <_  b  <->  ( 2  x.  (PellFund `  D )
)  <_  ( 2  x.  b ) ) )
5550, 44, 51, 53, 54syl112anc 1186 . . . . . . . . . . . . 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 ) ) )
5649, 55mpbid 201 . . . . . . . . . . . 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
) )
5737, 39, 46, 48, 56ltletrd 9066 . . . . . . . . . . 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 ) )
58 simp1 955 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  D  e.  ( NN  \NN ) )
59213ad2ant1 976 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  (Pell1QR `  D
)  C_  (Pell14QR `  D
) )
60 simp2l 981 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  e.  (Pell1QR `  D ) )
6159, 60sseldd 3257 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  e.  (Pell14QR `  D ) )
62 simp2r 982 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  (Pell1QR `  D ) )
6359, 62sseldd 3257 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  (Pell14QR `  D ) )
64 pell14qrdivcl 26273 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D )  /\  b  e.  (Pell14QR `  D )
)  ->  ( a  /  b )  e.  (Pell14QR `  D )
)
6558, 61, 63, 64syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( a  /  b )  e.  (Pell14QR `  D )
)
6658, 63, 43syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  RR )
6766recnd 8951 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  CC )
6867mulid2d 8943 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( 1  x.  b )  =  b )
69 simp3l 983 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  <  a )
7068, 69eqbrtrd 4124 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( 1  x.  b )  < 
a )
717a1i 10 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  1  e.  RR )
7258, 61, 23syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  e.  RR )
73 pell14qrgt0 26267 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( NN 
\NN )  /\  b  e.  (Pell14QR `  D ) )  -> 
0  <  b )
7458, 63, 73syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  0  <  b )
75 ltmuldiv 9716 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  RR  /\  a  e.  RR  /\  (
b  e.  RR  /\  0  <  b ) )  ->  ( ( 1  x.  b )  < 
a  <->  1  <  (
a  /  b ) ) )
7671, 72, 66, 74, 75syl112anc 1186 . . . . . . . . . . . . 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 ) ) )
7770, 76mpbid 201 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  1  <  ( a  /  b ) )
78 simp3r 984 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  <  ( 2  x.  b ) )
791a1i 10 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  2  e.  RR )
80 ltdivmul2 9721 . . . . . . . . . . . . . 14  |-  ( ( a  e.  RR  /\  2  e.  RR  /\  (
b  e.  RR  /\  0  <  b ) )  ->  ( ( a  /  b )  <  2  <->  a  <  (
2  x.  b ) ) )
8172, 79, 66, 74, 80syl112anc 1186 . . . . . . . . . . . . 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 ) ) )
8278, 81mpbird 223 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( a  /  b )  <  2 )
83 simprr 733 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  <  2
)
84 simpll 730 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  D  e.  ( NN  \NN ) )
85 simplr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  e.  (Pell14QR `  D ) )
86 simprl 732 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  1  <  (
a  /  b ) )
87 pell14qrgapw 26284 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  / 
b )  e.  (Pell14QR `  D )  /\  1  <  ( a  /  b
) )  ->  2  <  ( a  /  b
) )
8884, 85, 86, 87syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  2  <  (
a  /  b ) )
89 pell14qrre 26265 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  / 
b )  e.  (Pell14QR `  D ) )  -> 
( a  /  b
)  e.  RR )
9089adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  e.  RR )
91 ltnsym 9009 . . . . . . . . . . . . . . 15  |-  ( ( 2  e.  RR  /\  ( a  /  b
)  e.  RR )  ->  ( 2  < 
( a  /  b
)  ->  -.  (
a  /  b )  <  2 ) )
921, 90, 91sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( 2  < 
( a  /  b
)  ->  -.  (
a  /  b )  <  2 ) )
9388, 92mpd 14 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  -.  ( a  /  b )  <  2 )
94 pm2.24 101 . . . . . . . . . . . . 13  |-  ( ( a  /  b )  <  2  ->  ( -.  ( a  /  b
)  <  2  ->  (PellFund `  D )  e.  (Pell1QR `  D ) ) )
9583, 93, 94sylc 56 . . . . . . . . . . . 12  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
)
9658, 65, 77, 82, 95syl22anc 1183 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  (PellFund `  D
)  e.  (Pell1QR `  D
) )
9732, 34, 35, 36, 57, 96syl122anc 1191 . . . . . . . . . 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 )
)
9897ex 423 . . . . . . . . 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 )
) )
9998rexlimdva 2743 . . . . . . . 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 ) ) )
10030, 99mpd 14 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
)
101100exp32 588 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <  a  ->  ( a  <  ( 2  x.  (PellFund `  D ) )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
) ) )
102 simp2 956 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  =  a )
103 simp1r 980 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  a  e.  (Pell1QR `  D )
)
104102, 103eqeltrd 2432 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  e.  (Pell1QR `  D ) )
1051043exp 1150 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  =  a  ->  ( a  <  ( 2  x.  (PellFund `  D )
)  ->  (PellFund `  D
)  e.  (Pell1QR `  D
) ) ) )
106101, 105jaod 369 . . . . 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 ) ) ) )
10725, 106sylbid 206 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <_  a  ->  ( a  <  ( 2  x.  (PellFund `  D ) )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
) ) )
108107imp3a 420 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( ( (PellFund `  D
)  <_  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  e.  (Pell1QR `  D ) ) )
109108rexlimdva 2743 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( E. a  e.  (Pell1QR `  D )
( (PellFund `  D )  <_  a  /\  a  < 
( 2  x.  (PellFund `  D ) ) )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
) )
11019, 109mpd 14 1  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   E.wrex 2620    \ cdif 3225    C_ wss 3228   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   RRcr 8826   0cc0 8827   1c1 8828    + caddc 8830    x. cmul 8832    < clt 8957    <_ cle 8958    / cdiv 9513   NNcn 9836   2c2 9885  ◻NNcsquarenn 26244  Pell1QRcpell1qr 26245  Pell14QRcpell14qr 26247  PellFundcpellfund 26248
This theorem is referenced by:  pellfund14  26306  pellfund14b  26307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-omul 6571  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-acn 7665  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-q 10409  df-rp 10447  df-ico 10754  df-fz 10875  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-dvds 12629  df-gcd 12783  df-numer 12903  df-denom 12904  df-squarenn 26249  df-pell1qr 26250  df-pell14qr 26251  df-pell1234qr 26252  df-pellfund 26253
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