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Theorem pellfundex 26971
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 26961. (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 9815 . . . 4  |-  2  e.  RR
2 pellfundre 26966 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
3 remulcl 8822 . . . 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 8838 . . . . . . . 8  |-  0  e.  RR
65a1i 10 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
0  e.  RR )
7 1re 8837 . . . . . . . 8  |-  1  e.  RR
87a1i 10 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  RR )
9 0lt1 9296 . . . . . . . 8  |-  0  <  1
109a1i 10 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
0  <  1 )
11 pellfundgt1 26968 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
1  <  (PellFund `  D
) )
126, 8, 2, 10, 11lttrd 8977 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
0  <  (PellFund `  D
) )
132, 12elrpd 10388 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR+ )
142, 13ltaddrpd 10419 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  < 
( (PellFund `  D )  +  (PellFund `  D )
) )
152recnd 8861 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  CC )
16152timesd 9954 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( 2  x.  (PellFund `  D ) )  =  ( (PellFund `  D
)  +  (PellFund `  D
) ) )
1714, 16breqtrrd 4049 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  < 
( 2  x.  (PellFund `  D ) ) )
18 pellfundglb 26970 . . 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 26953 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
2221sselda 3180 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
a  e.  (Pell14QR `  D
) )
23 pell14qrre 26942 . . . . . . 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 8962 . . . . 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 26970 . . . . . . . . 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 3181 . . . . . . . . . . . . . 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 26942 . . . . . . . . . . . . . 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 8822 . . . . . . . . . . . . 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 9828 . . . . . . . . . . . . . . 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 9609 . . . . . . . . . . . . . 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 8976 . . . . . . . . . . 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 3181 . . . . . . . . . . . . 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 3181 . . . . . . . . . . . . 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 26950 . . . . . . . . . . . . 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 8861 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  CC )
6867mulid2d 8853 . . . . . . . . . . . . . 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 4043 . . . . . . . . . . . . 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 26944 . . . . . . . . . . . . . . 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 9626 . . . . . . . . . . . . . 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 9631 . . . . . . . . . . . . . 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 26961 . . . . . . . . . . . . . . 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 26942 . . . . . . . . . . . . . . . 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 8919 . . . . . . . . . . . . . . 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 2667 . . . . . . . 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 2357 . . . . . . 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 2667 . 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 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   2c2 9795  ◻NNcsquarenn 26921  Pell1QRcpell1qr 26922  Pell14QRcpell14qr 26924  PellFundcpellfund 26925
This theorem is referenced by:  pellfund14  26983  pellfund14b  26984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-ico 10662  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-numer 12806  df-denom 12807  df-squarenn 26926  df-pell1qr 26927  df-pell14qr 26928  df-pell1234qr 26929  df-pellfund 26930
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