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Theorem elpell1qr2 29216
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 29212 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
21sselda 3359 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  ->  A  e.  (Pell14QR `  D
) )
3 pell1qrge1 29214 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  -> 
1  <_  A )
42, 3jca 532 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  -> 
( A  e.  (Pell14QR `  D )  /\  1  <_  A ) )
5 1re 9388 . . . . . 6  |-  1  e.  RR
65a1i 11 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  e.  RR )
7 pell14qrre 29201 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
86, 7leloed 9520 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <_  A  <->  ( 1  <  A  \/  1  =  A )
) )
96, 7ltnled 9524 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <  A  <->  -.  A  <_  1 ) )
109biimpa 484 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  A  <_  1 )
11 1div1e1 10027 . . . . . . . . . . . . 13  |-  ( 1  /  1 )  =  1
1211eqcomi 2447 . . . . . . . . . . . 12  |-  1  =  ( 1  / 
1 )
1312a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  1  =  ( 1  /  1
) )
1413breq2d 4307 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  1  <->  A  <_  ( 1  /  1 ) ) )
157adantr 465 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  A  e.  RR )
16 pell14qrgt0 29203 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <  A )
1716adantr 465 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  0  <  A )
185a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  1  e.  RR )
19 0lt1 9865 . . . . . . . . . . . 12  |-  0  <  1
2019a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  0  <  1 )
21 lerec2 10223 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  e.  RR  /\  0  <  1 ) )  -> 
( A  <_  (
1  /  1 )  <->  1  <_  ( 1  /  A ) ) )
2215, 17, 18, 20, 21syl22anc 1219 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  ( 1  /  1
)  <->  1  <_  (
1  /  A ) ) )
2314, 22bitrd 253 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  1  <->  1  <_  (
1  /  A ) ) )
2410, 23mtbid 300 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  1  <_  ( 1  /  A
) )
25 simplll 757 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  /\  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  D  e.  ( NN  \NN ) )
26 pell1qrge1 29214 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  ( 1  /  A )  e.  (Pell1QR `  D ) )  -> 
1  <_  ( 1  /  A ) )
2725, 26sylancom 667 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  /\  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  1  <_  ( 1  /  A ) )
2824, 27mtand 659 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  (
1  /  A )  e.  (Pell1QR `  D
) )
29 pell14qrdich 29213 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  e.  (Pell1QR `  D )  \/  (
1  /  A )  e.  (Pell1QR `  D
) ) )
3029adantr 465 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  e.  (Pell1QR `  D )  \/  ( 1  /  A
)  e.  (Pell1QR `  D
) ) )
31 orel2 383 . . . . . . 7  |-  ( -.  ( 1  /  A
)  e.  (Pell1QR `  D
)  ->  ( ( A  e.  (Pell1QR `  D
)  \/  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  A  e.  (Pell1QR `  D ) ) )
3228, 30, 31sylc 60 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  A  e.  (Pell1QR `  D ) )
33 simpr 461 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  1  =  A )
34 pell1qr1 29215 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  (Pell1QR `  D
) )
3534ad2antrr 725 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  1  e.  (Pell1QR `  D )
)
3633, 35eqeltrrd 2518 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  A  e.  (Pell1QR `  D )
)
3732, 36jaodan 783 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  ( 1  <  A  \/  1  =  A ) )  ->  A  e.  (Pell1QR `  D ) )
3837ex 434 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( 1  < 
A  \/  1  =  A )  ->  A  e.  (Pell1QR `  D )
) )
398, 38sylbid 215 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <_  A  ->  A  e.  (Pell1QR `  D
) ) )
4039impr 619 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell14QR `  D )  /\  1  <_  A ) )  ->  A  e.  (Pell1QR `  D ) )
414, 40impbida 828 1  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1QR `  D )  <->  ( A  e.  (Pell14QR `  D )  /\  1  <_  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3328   class class class wbr 4295   ` cfv 5421  (class class class)co 6094   RRcr 9284   0cc0 9285   1c1 9286    < clt 9421    <_ cle 9422    / cdiv 9996   NNcn 10325  ◻NNcsquarenn 29180  Pell1QRcpell1qr 29181  Pell14QRcpell14qr 29183
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 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-2nd 6581  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-seq 11810  df-exp 11869  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-pell1qr 29186  df-pell14qr 29187  df-pell1234qr 29188
This theorem is referenced by:  pell14qrgap  29219  pellfundglb  29229
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