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Theorem pell14qrgapw 26961
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 9815 . . 3  |-  2  e.  RR
21a1i 10 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  e.  RR )
3 eldifi 3298 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
433ad2ant1 976 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  NN )
54nnrpd 10389 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  RR+ )
6 1rp 10358 . . . . . . 7  |-  1  e.  RR+
76a1i 10 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  e.  RR+ )
85, 7rpaddcld 10405 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( D  +  1 )  e.  RR+ )
98rpsqrcld 11894 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  ( D  + 
1 ) )  e.  RR+ )
109rpred 10390 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  ( D  + 
1 ) )  e.  RR )
115rpsqrcld 11894 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  D )  e.  RR+ )
1211rpred 10390 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  D )  e.  RR )
1310, 12readdcld 8862 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  e.  RR )
14 pell14qrre 26942 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
15143adant3 975 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  RR )
16 df-2 9804 . . 3  |-  2  =  ( 1  +  1 )
17 1re 8837 . . . . 5  |-  1  e.  RR
1817a1i 10 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  e.  RR )
194nnred 9761 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  RR )
20 peano2re 8985 . . . . . . . 8  |-  ( D  e.  RR  ->  ( D  +  1 )  e.  RR )
2119, 20syl 15 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( D  +  1 )  e.  RR )
224nnge1d 9788 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <_  D )
2319ltp1d 9687 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  <  ( D  +  1 ) )
2418, 19, 21, 22, 23lelttrd 8974 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  ( D  +  1 ) )
25 sq1 11198 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2625a1i 10 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  =  1 )
274nncnd 9762 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  CC )
28 peano2cn 8984 . . . . . . . 8  |-  ( D  e.  CC  ->  ( D  +  1 )  e.  CC )
2927, 28syl 15 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( D  +  1 )  e.  CC )
3029sqsqrd 11921 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) ) ^ 2 )  =  ( D  + 
1 ) )
3124, 26, 303brtr4d 4053 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  <  ( ( sqr `  ( D  +  1 ) ) ^ 2 ) )
32 0le1 9297 . . . . . . 7  |-  0  <_  1
3332a1i 10 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  1 )
349rpge0d 10394 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  ( sqr `  ( D  +  1 ) ) )
3518, 10, 33, 34lt2sqd 11279 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1  <  ( sqr `  ( D  +  1 ) )  <->  ( 1 ^ 2 )  < 
( ( sqr `  ( D  +  1 ) ) ^ 2 ) ) )
3631, 35mpbird 223 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  ( sqr `  ( D  +  1 ) ) )
3727sqsqrd 11921 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  D
) ^ 2 )  =  D )
3822, 26, 373brtr4d 4053 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  <_  ( ( sqr `  D ) ^ 2 ) )
3911rpge0d 10394 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  ( sqr `  D
) )
4018, 12, 33, 39le2sqd 11280 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1  <_  ( sqr `  D )  <->  ( 1 ^ 2 )  <_ 
( ( sqr `  D
) ^ 2 ) ) )
4138, 40mpbird 223 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <_  ( sqr `  D
) )
4218, 18, 10, 12, 36, 41ltleaddd 9392 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1  +  1 )  <  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) ) )
4316, 42syl5eqbr 4056 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  <  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) ) )
44 pell14qrgap 26960 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A )
452, 13, 15, 43, 44ltletrd 8976 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868   NNcn 9746   2c2 9795   RR+crp 10354   ^cexp 11104   sqrcsqr 11718  ◻NNcsquarenn 26921  Pell14QRcpell14qr 26924
This theorem is referenced by:  pellfundex  26971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-pell1qr 26927  df-pell14qr 26928  df-pell1234qr 26929
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