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Theorem pellfundex 35653
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 35641. (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 10679 . . . 4  |-  2  e.  RR
2 pellfundre 35648 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
3 remulcl 9624 . . . 4  |-  ( ( 2  e.  RR  /\  (PellFund `  D )  e.  RR )  ->  (
2  x.  (PellFund `  D
) )  e.  RR )
41, 2, 3sylancr 667 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( 2  x.  (PellFund `  D ) )  e.  RR )
5 0red 9644 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
0  e.  RR )
6 1red 9658 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  RR )
7 0lt1 10136 . . . . . . . 8  |-  0  <  1
87a1i 11 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
0  <  1 )
9 pellfundgt1 35650 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
1  <  (PellFund `  D
) )
105, 6, 2, 8, 9lttrd 9796 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
0  <  (PellFund `  D
) )
112, 10elrpd 11338 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR+ )
122, 11ltaddrpd 11371 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  < 
( (PellFund `  D )  +  (PellFund `  D )
) )
132recnd 9669 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  CC )
14132timesd 10855 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( 2  x.  (PellFund `  D ) )  =  ( (PellFund `  D
)  +  (PellFund `  D
) ) )
1512, 14breqtrrd 4447 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  < 
( 2  x.  (PellFund `  D ) ) )
16 pellfundglb 35652 . . 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
) ) ) )
174, 15, 16mpd3an23 1362 . 2  |-  ( D  e.  ( NN  \NN )  ->  E. a  e.  (Pell1QR `  D ) ( (PellFund `  D )  <_  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )
182adantr 466 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
(PellFund `  D )  e.  RR )
19 pell1qrss14 35633 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
2019sselda 3464 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
a  e.  (Pell14QR `  D
) )
21 pell14qrre 35622 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D ) )  -> 
a  e.  RR )
2220, 21syldan 472 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
a  e.  RR )
2318, 22leloed 9778 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <_  a  <->  ( (PellFund `  D
)  <  a  \/  (PellFund `  D )  =  a ) ) )
24 simp-4l 774 . . . . . . . . 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 ) )  ->  D  e.  ( NN  \NN )
)
25 simp-4r 775 . . . . . . . . 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 ) )  -> 
a  e.  (Pell1QR `  D
) )
26 simplr 760 . . . . . . . . 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 ) )  -> 
b  e.  (Pell1QR `  D
) )
27 simprr 764 . . . . . . . . 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 ) )  -> 
b  <  a )
2822ad3antrrr 734 . . . . . . . . . 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 ) )  -> 
a  e.  RR )
294ad4antr 736 . . . . . . . . . 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 ) )  -> 
( 2  x.  (PellFund `  D ) )  e.  RR )
3019ad4antr 736 . . . . . . . . . . . . 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 ) )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
3130, 26sseldd 3465 . . . . . . . . . . . 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 ) )  -> 
b  e.  (Pell14QR `  D
) )
32 pell14qrre 35622 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  b  e.  (Pell14QR `  D ) )  -> 
b  e.  RR )
3324, 31, 32syl2anc 665 . . . . . . . . . . 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.  RR )
34 remulcl 9624 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\  b  e.  RR )  ->  ( 2  x.  b
)  e.  RR )
351, 33, 34sylancr 667 . . . . . . . . . 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 ) )  -> 
( 2  x.  b
)  e.  RR )
36 simprr 764 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  a  <  (
2  x.  (PellFund `  D
) ) )
3736ad2antrr 730 . . . . . . . . . 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 ) )  -> 
a  <  ( 2  x.  (PellFund `  D
) ) )
38 simprl 762 . . . . . . . . . . 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 ) )  -> 
(PellFund `  D )  <_ 
b )
392ad4antr 736 . . . . . . . . . . . 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 ) )  -> 
(PellFund `  D )  e.  RR )
401a1i 11 . . . . . . . . . . . 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  e.  RR )
41 2pos 10701 . . . . . . . . . . . . 13  |-  0  <  2
4241a1i 11 . . . . . . . . . . . 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 ) )  -> 
0  <  2 )
43 lemul2 10458 . . . . . . . . . . . 12  |-  ( ( (PellFund `  D )  e.  RR  /\  b  e.  RR  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
( (PellFund `  D )  <_  b  <->  ( 2  x.  (PellFund `  D )
)  <_  ( 2  x.  b ) ) )
4439, 33, 40, 42, 43syl112anc 1268 . . . . . . . . . . 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 ) )  -> 
( (PellFund `  D )  <_  b  <->  ( 2  x.  (PellFund `  D )
)  <_  ( 2  x.  b ) ) )
4538, 44mpbid 213 . . . . . . . . . 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 ) )  -> 
( 2  x.  (PellFund `  D ) )  <_ 
( 2  x.  b
) )
4628, 29, 35, 37, 45ltletrd 9795 . . . . . . . . 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 ) )  -> 
a  <  ( 2  x.  b ) )
47 simp1 1005 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  D  e.  ( NN  \NN ) )
48193ad2ant1 1026 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  (Pell1QR `  D
)  C_  (Pell14QR `  D
) )
49 simp2l 1031 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  e.  (Pell1QR `  D ) )
5048, 49sseldd 3465 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  e.  (Pell14QR `  D ) )
51 simp2r 1032 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  (Pell1QR `  D ) )
5248, 51sseldd 3465 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  (Pell14QR `  D ) )
53 pell14qrdivcl 35630 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell14QR `  D )  /\  b  e.  (Pell14QR `  D )
)  ->  ( a  /  b )  e.  (Pell14QR `  D )
)
5447, 50, 52, 53syl3anc 1264 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( a  /  b )  e.  (Pell14QR `  D )
)
5547, 52, 32syl2anc 665 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  RR )
5655recnd 9669 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  e.  CC )
5756mulid2d 9661 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( 1  x.  b )  =  b )
58 simp3l 1033 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  b  <  a )
5957, 58eqbrtrd 4441 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( 1  x.  b )  < 
a )
60 1red 9658 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  1  e.  RR )
6147, 50, 21syl2anc 665 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  e.  RR )
62 pell14qrgt0 35624 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  b  e.  (Pell14QR `  D ) )  -> 
0  <  b )
6347, 52, 62syl2anc 665 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  0  <  b )
64 ltmuldiv 10478 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  a  e.  RR  /\  (
b  e.  RR  /\  0  <  b ) )  ->  ( ( 1  x.  b )  < 
a  <->  1  <  (
a  /  b ) ) )
6560, 61, 55, 63, 64syl112anc 1268 . . . . . . . . . . 11  |-  ( ( 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 ) ) )
6659, 65mpbid 213 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  1  <  ( a  /  b ) )
67 simp3r 1034 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  a  <  ( 2  x.  b ) )
681a1i 11 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  2  e.  RR )
69 ltdivmul2 10482 . . . . . . . . . . . 12  |-  ( ( a  e.  RR  /\  2  e.  RR  /\  (
b  e.  RR  /\  0  <  b ) )  ->  ( ( a  /  b )  <  2  <->  a  <  (
2  x.  b ) ) )
7061, 68, 55, 63, 69syl112anc 1268 . . . . . . . . . . 11  |-  ( ( 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 ) ) )
7167, 70mpbird 235 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  ( a  /  b )  <  2 )
72 simprr 764 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  <  2
)
73 simpll 758 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  D  e.  ( NN  \NN ) )
74 simplr 760 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  e.  (Pell14QR `  D ) )
75 simprl 762 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  1  <  (
a  /  b ) )
76 pell14qrgapw 35641 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  / 
b )  e.  (Pell14QR `  D )  /\  1  <  ( a  /  b
) )  ->  2  <  ( a  /  b
) )
7773, 74, 75, 76syl3anc 1264 . . . . . . . . . . . 12  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  2  <  (
a  /  b ) )
78 pell14qrre 35622 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  / 
b )  e.  (Pell14QR `  D ) )  -> 
( a  /  b
)  e.  RR )
7978adantr 466 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( a  / 
b )  e.  RR )
80 ltnsym 9732 . . . . . . . . . . . . 13  |-  ( ( 2  e.  RR  /\  ( a  /  b
)  e.  RR )  ->  ( 2  < 
( a  /  b
)  ->  -.  (
a  /  b )  <  2 ) )
811, 79, 80sylancr 667 . . . . . . . . . . . 12  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  ( 2  < 
( a  /  b
)  ->  -.  (
a  /  b )  <  2 ) )
8277, 81mpd 15 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  -.  ( a  /  b )  <  2 )
8372, 82pm2.21dd 177 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  ( a  /  b )  e.  (Pell14QR `  D )
)  /\  ( 1  <  ( a  / 
b )  /\  (
a  /  b )  <  2 ) )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
)
8447, 54, 66, 71, 83syl22anc 1265 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  ( a  e.  (Pell1QR `  D )  /\  b  e.  (Pell1QR `  D ) )  /\  ( b  <  a  /\  a  <  ( 2  x.  b ) ) )  ->  (PellFund `  D
)  e.  (Pell1QR `  D
) )
8524, 25, 26, 27, 46, 84syl122anc 1273 . . . . . . . 8  |-  ( ( ( ( ( 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 )
)
86 simpll 758 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  D  e.  ( NN  \NN ) )
8722adantr 466 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  a  e.  RR )
88 simprl 762 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  (PellFund `  D )  <  a )
89 pellfundglb 35652 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  RR  /\  (PellFund `  D )  <  a )  ->  E. b  e.  (Pell1QR `  D )
( (PellFund `  D )  <_  b  /\  b  < 
a ) )
9086, 87, 88, 89syl3anc 1264 . . . . . . . 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 ) )
9185, 90r19.29a 2970 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  ( (PellFund `  D )  <  a  /\  a  <  ( 2  x.  (PellFund `  D
) ) ) )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
)
9291exp32 608 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <  a  ->  ( a  <  ( 2  x.  (PellFund `  D ) )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
) ) )
93 simp2 1006 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  =  a )
94 simp1r 1030 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  a  e.  (Pell1QR `  D )
)
9593, 94eqeltrd 2510 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  a  e.  (Pell1QR `  D )
)  /\  (PellFund `  D
)  =  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  e.  (Pell1QR `  D ) )
96953exp 1204 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  =  a  ->  ( a  <  ( 2  x.  (PellFund `  D )
)  ->  (PellFund `  D
)  e.  (Pell1QR `  D
) ) ) )
9792, 96jaod 381 . . . . 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 ) ) ) )
9823, 97sylbid 218 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( (PellFund `  D )  <_  a  ->  ( a  <  ( 2  x.  (PellFund `  D ) )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
) ) )
9998impd 432 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  a  e.  (Pell1QR `  D ) )  -> 
( ( (PellFund `  D
)  <_  a  /\  a  <  ( 2  x.  (PellFund `  D )
) )  ->  (PellFund `  D )  e.  (Pell1QR `  D ) ) )
10099rexlimdva 2917 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( E. a  e.  (Pell1QR `  D )
( (PellFund `  D )  <_  a  /\  a  < 
( 2  x.  (PellFund `  D ) ) )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
) )
10117, 100mpd 15 1  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  (Pell1QR `  D )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   E.wrex 2776    \ cdif 3433    C_ wss 3436   class class class wbr 4420   ` cfv 5597  (class class class)co 6301   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    / cdiv 10269   NNcn 10609   2c2 10659  ◻NNcsquarenn 35599  Pell1QRcpell1qr 35600  Pell14QRcpell14qr 35602  PellFundcpellfund 35603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-oadd 7190  df-omul 7191  df-er 7367  df-map 7478  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-acn 8377  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-ico 11641  df-fz 11785  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-hash 12515  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-dvds 14293  df-gcd 14456  df-numer 14671  df-denom 14672  df-squarenn 35605  df-pell1qr 35606  df-pell14qr 35607  df-pell1234qr 35608  df-pellfund 35609
This theorem is referenced by:  pellfund14  35665  pellfund14b  35666
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