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Theorem elpell1qr2 26957
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 26953 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
21sselda 3180 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  ->  A  e.  (Pell14QR `  D
) )
3 pell1qrge1 26955 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  -> 
1  <_  A )
42, 3jca 518 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  -> 
( A  e.  (Pell14QR `  D )  /\  1  <_  A ) )
5 1re 8837 . . . . . 6  |-  1  e.  RR
65a1i 10 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  e.  RR )
7 pell14qrre 26942 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
86, 7leloed 8962 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <_  A  <->  ( 1  <  A  \/  1  =  A )
) )
96, 7ltnled 8966 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <  A  <->  -.  A  <_  1 ) )
109biimpa 470 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  A  <_  1 )
11 ax-1cn 8795 . . . . . . . . . . . . . 14  |-  1  e.  CC
1211div1i 9488 . . . . . . . . . . . . 13  |-  ( 1  /  1 )  =  1
1312eqcomi 2287 . . . . . . . . . . . 12  |-  1  =  ( 1  / 
1 )
1413a1i 10 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  1  =  ( 1  /  1
) )
1514breq2d 4035 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  1  <->  A  <_  ( 1  /  1 ) ) )
167adantr 451 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  A  e.  RR )
17 pell14qrgt0 26944 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <  A )
1817adantr 451 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  0  <  A )
195a1i 10 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  1  e.  RR )
20 0lt1 9296 . . . . . . . . . . . 12  |-  0  <  1
2120a1i 10 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  0  <  1 )
22 lerec2 9644 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  e.  RR  /\  0  <  1 ) )  -> 
( A  <_  (
1  /  1 )  <->  1  <_  ( 1  /  A ) ) )
2316, 18, 19, 21, 22syl22anc 1183 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  ( 1  /  1
)  <->  1  <_  (
1  /  A ) ) )
2415, 23bitrd 244 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  1  <->  1  <_  (
1  /  A ) ) )
2510, 24mtbid 291 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  1  <_  ( 1  /  A
) )
26 simplll 734 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  /\  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  D  e.  ( NN  \NN ) )
27 pell1qrge1 26955 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  ( 1  /  A )  e.  (Pell1QR `  D ) )  -> 
1  <_  ( 1  /  A ) )
2826, 27sylancom 648 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  /\  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  1  <_  ( 1  /  A ) )
2925, 28mtand 640 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  (
1  /  A )  e.  (Pell1QR `  D
) )
30 pell14qrdich 26954 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  e.  (Pell1QR `  D )  \/  (
1  /  A )  e.  (Pell1QR `  D
) ) )
3130adantr 451 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  e.  (Pell1QR `  D )  \/  ( 1  /  A
)  e.  (Pell1QR `  D
) ) )
32 orel2 372 . . . . . . 7  |-  ( -.  ( 1  /  A
)  e.  (Pell1QR `  D
)  ->  ( ( A  e.  (Pell1QR `  D
)  \/  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  A  e.  (Pell1QR `  D ) ) )
3329, 31, 32sylc 56 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  A  e.  (Pell1QR `  D ) )
34 simpr 447 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  1  =  A )
35 pell1qr1 26956 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  (Pell1QR `  D
) )
3635ad2antrr 706 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  1  e.  (Pell1QR `  D )
)
3734, 36eqeltrrd 2358 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  A  e.  (Pell1QR `  D )
)
3833, 37jaodan 760 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  ( 1  <  A  \/  1  =  A ) )  ->  A  e.  (Pell1QR `  D ) )
3938ex 423 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( 1  < 
A  \/  1  =  A )  ->  A  e.  (Pell1QR `  D )
) )
408, 39sylbid 206 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <_  A  ->  A  e.  (Pell1QR `  D
) ) )
4140impr 602 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell14QR `  D )  /\  1  <_  A ) )  ->  A  e.  (Pell1QR `  D ) )
424, 41impbida 805 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746  ◻NNcsquarenn 26921  Pell1QRcpell1qr 26922  Pell14QRcpell14qr 26924
This theorem is referenced by:  pell14qrgap  26960  pellfundglb  26970
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  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-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-pell1qr 26927  df-pell14qr 26928  df-pell1234qr 26929
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