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Theorem elpell1qr2 30412
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 30408 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
21sselda 3504 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  ->  A  e.  (Pell14QR `  D
) )
3 pell1qrge1 30410 . . 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 9591 . . . . . 6  |-  1  e.  RR
65a1i 11 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  e.  RR )
7 pell14qrre 30397 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
86, 7leloed 9723 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <_  A  <->  ( 1  <  A  \/  1  =  A )
) )
96, 7ltnled 9727 . . . . . . . . . 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 10233 . . . . . . . . . . . . 13  |-  ( 1  /  1 )  =  1
1211eqcomi 2480 . . . . . . . . . . . 12  |-  1  =  ( 1  / 
1 )
1312a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  1  =  ( 1  /  1
) )
1413breq2d 4459 . . . . . . . . . 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 30399 . . . . . . . . . . . 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 10071 . . . . . . . . . . . 12  |-  0  <  1
2019a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  0  <  1 )
21 lerec2 10429 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  e.  RR  /\  0  <  1 ) )  -> 
( A  <_  (
1  /  1 )  <->  1  <_  ( 1  /  A ) ) )
2215, 17, 18, 20, 21syl22anc 1229 . . . . . . . . . 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 30410 . . . . . . . . 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 30409 . . . . . . . 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 30411 . . . . . . . 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 2556 . . . . . 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 830 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 1379    e. wcel 1767    \ cdif 3473   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   RRcr 9487   0cc0 9488   1c1 9489    < clt 9624    <_ cle 9625    / cdiv 10202   NNcn 10532  ◻NNcsquarenn 30376  Pell1QRcpell1qr 30377  Pell14QRcpell14qr 30379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-pell1qr 30382  df-pell14qr 30383  df-pell1234qr 30384
This theorem is referenced by:  pell14qrgap  30415  pellfundglb  30425
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