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Theorem elpell1qr2 31050
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 31046 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
21sselda 3489 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  ->  A  e.  (Pell14QR `  D
) )
3 pell1qrge1 31048 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  -> 
1  <_  A )
42, 3jca 530 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D ) )  -> 
( A  e.  (Pell14QR `  D )  /\  1  <_  A ) )
5 1red 9600 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
1  e.  RR )
6 pell14qrre 31035 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
75, 6leloed 9717 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <_  A  <->  ( 1  <  A  \/  1  =  A )
) )
85, 6ltnled 9721 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <  A  <->  -.  A  <_  1 ) )
98biimpa 482 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  A  <_  1 )
10 1div1e1 10233 . . . . . . . . . . . . 13  |-  ( 1  /  1 )  =  1
1110eqcomi 2467 . . . . . . . . . . . 12  |-  1  =  ( 1  / 
1 )
1211a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  1  =  ( 1  /  1
) )
1312breq2d 4451 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  1  <->  A  <_  ( 1  /  1 ) ) )
146adantr 463 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  A  e.  RR )
15 pell14qrgt0 31037 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
0  <  A )
1615adantr 463 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  0  <  A )
17 1red 9600 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  1  e.  RR )
18 0lt1 10071 . . . . . . . . . . . 12  |-  0  <  1
1918a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  0  <  1 )
20 lerec2 10428 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  e.  RR  /\  0  <  1 ) )  -> 
( A  <_  (
1  /  1 )  <->  1  <_  ( 1  /  A ) ) )
2114, 16, 17, 19, 20syl22anc 1227 . . . . . . . . . 10  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  ( 1  /  1
)  <->  1  <_  (
1  /  A ) ) )
2213, 21bitrd 253 . . . . . . . . 9  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  <_  1  <->  1  <_  (
1  /  A ) ) )
239, 22mtbid 298 . . . . . . . 8  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  1  <_  ( 1  /  A
) )
24 simplll 757 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  /\  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  D  e.  ( NN  \NN ) )
25 pell1qrge1 31048 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  ( 1  /  A )  e.  (Pell1QR `  D ) )  -> 
1  <_  ( 1  /  A ) )
2624, 25sylancom 665 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  /\  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  1  <_  ( 1  /  A ) )
2723, 26mtand 657 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  -.  (
1  /  A )  e.  (Pell1QR `  D
) )
28 pell14qrdich 31047 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  e.  (Pell1QR `  D )  \/  (
1  /  A )  e.  (Pell1QR `  D
) ) )
2928adantr 463 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  ( A  e.  (Pell1QR `  D )  \/  ( 1  /  A
)  e.  (Pell1QR `  D
) ) )
30 orel2 381 . . . . . . 7  |-  ( -.  ( 1  /  A
)  e.  (Pell1QR `  D
)  ->  ( ( A  e.  (Pell1QR `  D
)  \/  ( 1  /  A )  e.  (Pell1QR `  D )
)  ->  A  e.  (Pell1QR `  D ) ) )
3127, 29, 30sylc 60 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  <  A )  ->  A  e.  (Pell1QR `  D ) )
32 simpr 459 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  1  =  A )
33 pell1qr1 31049 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  -> 
1  e.  (Pell1QR `  D
) )
3433ad2antrr 723 . . . . . . 7  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  1  e.  (Pell1QR `  D )
)
3532, 34eqeltrrd 2543 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  1  =  A )  ->  A  e.  (Pell1QR `  D )
)
3631, 35jaodan 783 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  ( 1  <  A  \/  1  =  A ) )  ->  A  e.  (Pell1QR `  D ) )
3736ex 432 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( 1  < 
A  \/  1  =  A )  ->  A  e.  (Pell1QR `  D )
) )
387, 37sylbid 215 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  <_  A  ->  A  e.  (Pell1QR `  D
) ) )
3938impr 617 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  ( A  e.  (Pell14QR `  D )  /\  1  <_  A ) )  ->  A  e.  (Pell1QR `  D ) )
404, 39impbida 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 366    /\ wa 367    = wceq 1398    e. wcel 1823    \ cdif 3458   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    < clt 9617    <_ cle 9618    / cdiv 10202   NNcn 10531  ◻NNcsquarenn 31014  Pell1QRcpell1qr 31015  Pell14QRcpell14qr 31017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-seq 12093  df-exp 12152  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-pell1qr 31020  df-pell14qr 31021  df-pell1234qr 31022
This theorem is referenced by:  pell14qrgap  31053  pellfundglb  31063
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