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Theorem elpell1qr2 26935
Description: The first quadrant solutions are precisely the positive Pell solutions which are at least one. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
elpell1qr2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1QR `  D )  <->  ( A  e.  (Pell14QR `  D )  /\  1  <_  A ) ) )

Proof of Theorem elpell1qr2
StepHypRef Expression
1 pell1qrss14 26931 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
21sselda 3348 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  ->  A  e.  (Pell14QR `  D
) )
3 pell1qrge1 26933 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  -> 
1  <_  A )
42, 3jca 519 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  -> 
( A  e.  (Pell14QR `  D )  /\  1  <_  A ) )
5 1re 9090 . . . . . 6  |-  1  e.  RR
65a1i 11 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  e.  RR )
7 pell14qrre 26920 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
86, 7leloed 9216 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <_  A  <->  ( 1  <  A  \/  1  =  A )
) )
96, 7ltnled 9220 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <  A  <->  -.  A  <_  1 ) )
109biimpa 471 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  A  <_  1 )
11 ax-1cn 9048 . . . . . . . . . . . . . 14  |-  1  e.  CC
1211div1i 9742 . . . . . . . . . . . . 13  |-  ( 1  /  1 )  =  1
1312eqcomi 2440 . . . . . . . . . . . 12  |-  1  =  ( 1  / 
1 )
1413a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  1  =  ( 1  /  1
) )
1514breq2d 4224 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  1  <->  A  <_  ( 1  /  1 ) ) )
167adantr 452 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  A  e.  RR )
17 pell14qrgt0 26922 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <  A )
1817adantr 452 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  0  <  A )
195a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  1  e.  RR )
20 0lt1 9550 . . . . . . . . . . . 12  |-  0  <  1
2120a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  0  <  1 )
22 lerec2 9898 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  e.  RR  /\  0  <  1 ) )  -> 
( A  <_  (
1  /  1 )  <->  1  <_  ( 1  /  A ) ) )
2316, 18, 19, 21, 22syl22anc 1185 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  ( 1  /  1
)  <->  1  <_  (
1  /  A ) ) )
2415, 23bitrd 245 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  1  <->  1  <_  (
1  /  A ) ) )
2510, 24mtbid 292 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  1  <_  ( 1  /  A
) )
26 simplll 735 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  /\  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  D  e.  ( NN  \NN ) )
27 pell1qrge1 26933 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  ( 1  /  A )  e.  (Pell1QR `  D ) )  -> 
1  <_  ( 1  /  A ) )
2826, 27sylancom 649 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  /\  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  1  <_  ( 1  /  A ) )
2925, 28mtand 641 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  (
1  /  A )  e.  (Pell1QR `  D
) )
30 pell14qrdich 26932 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  e.  (Pell1QR `  D )  \/  (
1  /  A )  e.  (Pell1QR `  D
) ) )
3130adantr 452 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  e.  (Pell1QR `  D )  \/  ( 1  /  A
)  e.  (Pell1QR `  D
) ) )
32 orel2 373 . . . . . . 7  |-  ( -.  ( 1  /  A
)  e.  (Pell1QR `  D
)  ->  ( ( A  e.  (Pell1QR `  D
)  \/  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  A  e.  (Pell1QR `  D ) ) )
3329, 31, 32sylc 58 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  A  e.  (Pell1QR `  D ) )
34 simpr 448 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  1  =  A )
35 pell1qr1 26934 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  (Pell1QR `  D
) )
3635ad2antrr 707 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  1  e.  (Pell1QR `  D )
)
3734, 36eqeltrrd 2511 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  A  e.  (Pell1QR `  D )
)
3833, 37jaodan 761 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  ( 1  <  A  \/  1  =  A ) )  ->  A  e.  (Pell1QR `  D ) )
3938ex 424 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( 1  < 
A  \/  1  =  A )  ->  A  e.  (Pell1QR `  D )
) )
408, 39sylbid 207 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <_  A  ->  A  e.  (Pell1QR `  D
) ) )
4140impr 603 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell14QR `  D )  /\  1  <_  A ) )  ->  A  e.  (Pell1QR `  D ) )
424, 41impbida 806 1  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1QR `  D )  <->  ( A  e.  (Pell14QR `  D )  /\  1  <_  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3317   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991    < clt 9120    <_ cle 9121    / cdiv 9677   NNcn 10000  ◻NNcsquarenn 26899  Pell1QRcpell1qr 26900  Pell14QRcpell14qr 26902
This theorem is referenced by:  pell14qrgap  26938  pellfundglb  26948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-pell1qr 26905  df-pell14qr 26906  df-pell1234qr 26907
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