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Theorem pell14qrgapw 26930
Description: Positive Pell solutions are bounded away from 1, with a friendlier bound. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell14qrgapw  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  <  A )

Proof of Theorem pell14qrgapw
StepHypRef Expression
1 2re 10061 . . 3  |-  2  e.  RR
21a1i 11 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  e.  RR )
3 eldifi 3461 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
433ad2ant1 978 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  NN )
54nnrpd 10639 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  RR+ )
6 1rp 10608 . . . . . . 7  |-  1  e.  RR+
76a1i 11 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  e.  RR+ )
85, 7rpaddcld 10655 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( D  +  1 )  e.  RR+ )
98rpsqrcld 12206 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  ( D  + 
1 ) )  e.  RR+ )
109rpred 10640 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  ( D  + 
1 ) )  e.  RR )
115rpsqrcld 12206 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  D )  e.  RR+ )
1211rpred 10640 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  D )  e.  RR )
1310, 12readdcld 9107 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  e.  RR )
14 pell14qrre 26911 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
15143adant3 977 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  RR )
16 df-2 10050 . . 3  |-  2  =  ( 1  +  1 )
17 1re 9082 . . . . 5  |-  1  e.  RR
1817a1i 11 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  e.  RR )
194nnred 10007 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  RR )
20 peano2re 9231 . . . . . . . 8  |-  ( D  e.  RR  ->  ( D  +  1 )  e.  RR )
2119, 20syl 16 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( D  +  1 )  e.  RR )
224nnge1d 10034 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <_  D )
2319ltp1d 9933 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  <  ( D  +  1 ) )
2418, 19, 21, 22, 23lelttrd 9220 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  ( D  +  1 ) )
25 sq1 11468 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2625a1i 11 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  =  1 )
274nncnd 10008 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  CC )
28 peano2cn 9230 . . . . . . . 8  |-  ( D  e.  CC  ->  ( D  +  1 )  e.  CC )
2927, 28syl 16 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( D  +  1 )  e.  CC )
3029sqsqrd 12233 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) ) ^ 2 )  =  ( D  + 
1 ) )
3124, 26, 303brtr4d 4234 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  <  ( ( sqr `  ( D  +  1 ) ) ^ 2 ) )
32 0le1 9543 . . . . . . 7  |-  0  <_  1
3332a1i 11 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  1 )
349rpge0d 10644 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  ( sqr `  ( D  +  1 ) ) )
3518, 10, 33, 34lt2sqd 11549 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1  <  ( sqr `  ( D  +  1 ) )  <->  ( 1 ^ 2 )  < 
( ( sqr `  ( D  +  1 ) ) ^ 2 ) ) )
3631, 35mpbird 224 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  ( sqr `  ( D  +  1 ) ) )
3727sqsqrd 12233 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  D
) ^ 2 )  =  D )
3822, 26, 373brtr4d 4234 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  <_  ( ( sqr `  D ) ^ 2 ) )
3911rpge0d 10644 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  ( sqr `  D
) )
4018, 12, 33, 39le2sqd 11550 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1  <_  ( sqr `  D )  <->  ( 1 ^ 2 )  <_ 
( ( sqr `  D
) ^ 2 ) ) )
4138, 40mpbird 224 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <_  ( sqr `  D
) )
4218, 18, 10, 12, 36, 41ltleaddd 9638 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1  +  1 )  <  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) ) )
4316, 42syl5eqbr 4237 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  <  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) ) )
44 pell14qrgap 26929 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A )
452, 13, 15, 43, 44ltletrd 9222 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3309   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    < clt 9112    <_ cle 9113   NNcn 9992   2c2 10041   RR+crp 10604   ^cexp 11374   sqrcsqr 12030  ◻NNcsquarenn 26890  Pell14QRcpell14qr 26893
This theorem is referenced by:  pellfundex  26940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-pell1qr 26896  df-pell14qr 26897  df-pell1234qr 26898
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