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Theorem elpell14qr2 29201
Description: A number is a positive Pell solution iff it is positive and a Pell solution, justifying our name choice. (Contributed by Stefan O'Rear, 19-Oct-2014.)
Assertion
Ref Expression
elpell14qr2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell14QR `  D )  <->  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) ) )

Proof of Theorem elpell14qr2
StepHypRef Expression
1 pell14qrss1234 29195 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  (Pell1234QR `
 D ) )
21sselda 3355 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  (Pell1234QR `  D
) )
3 pell14qrgt0 29198 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <  A )
42, 3jca 532 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  e.  (Pell1234QR `  D )  /\  0  <  A ) )
5 0re 9385 . . . . . . 7  |-  0  e.  RR
6 pell1234qrre 29191 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  RR )
7 ltnsym 9472 . . . . . . 7  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  -.  A  <  0
) )
85, 6, 7sylancr 663 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( 0  <  A  ->  -.  A  <  0
) )
98impr 619 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) )  ->  -.  A  <  0 )
106adantrr 716 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) )  ->  A  e.  RR )
1110lt0neg1d 9908 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) )  ->  ( A  <  0  <->  0  <  -u A
) )
129, 11mtbid 300 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) )  ->  -.  0  <  -u A )
13 pell14qrgt0 29198 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  -u A  e.  (Pell14QR `  D ) )  -> 
0  <  -u A )
1413ex 434 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( -u A  e.  (Pell14QR `  D )  ->  0  <  -u A ) )
1514adantr 465 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) )  ->  ( -u A  e.  (Pell14QR `  D )  ->  0  <  -u A
) )
1612, 15mtod 177 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) )  ->  -.  -u A  e.  (Pell14QR `  D )
)
17 pell1234qrdich 29200 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D )
) )
1817adantrr 716 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) )  ->  ( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D ) ) )
19 orel2 383 . . 3  |-  ( -.  -u A  e.  (Pell14QR `  D )  ->  (
( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D )
)  ->  A  e.  (Pell14QR `  D ) ) )
2016, 18, 19sylc 60 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) )  ->  A  e.  (Pell14QR `  D ) )
214, 20impbida 828 1  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell14QR `  D )  <->  ( A  e.  (Pell1234QR `  D )  /\  0  <  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    e. wcel 1756    \ cdif 3324   class class class wbr 4291   ` cfv 5417   RRcr 9280   0cc0 9281    < clt 9417   -ucneg 9595   NNcn 10321  ◻NNcsquarenn 29175  Pell1234QRcpell1234qr 29177  Pell14QRcpell14qr 29178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-pell14qr 29182  df-pell1234qr 29183
This theorem is referenced by:  pell14qrmulcl  29202  pell14qrreccl  29203
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