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Theorem pell14qrgapw 35722
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 10679 . . 3  |-  2  e.  RR
21a1i 11 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  e.  RR )
3 eldifi 3555 . . . . . . . 8  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
433ad2ant1 1029 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  NN )
54nnrpd 11339 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  RR+ )
6 1rp 11306 . . . . . . 7  |-  1  e.  RR+
76a1i 11 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  e.  RR+ )
85, 7rpaddcld 11356 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( D  +  1 )  e.  RR+ )
98rpsqrtcld 13473 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  ( D  + 
1 ) )  e.  RR+ )
109rpred 11341 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  ( D  + 
1 ) )  e.  RR )
115rpsqrtcld 13473 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  D )  e.  RR+ )
1211rpred 11341 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  D )  e.  RR )
1310, 12readdcld 9670 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  e.  RR )
14 pell14qrre 35703 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
15143adant3 1028 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  RR )
16 df-2 10668 . . 3  |-  2  =  ( 1  +  1 )
17 1red 9658 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  e.  RR )
184nnred 10624 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  RR )
19 peano2re 9806 . . . . . . . 8  |-  ( D  e.  RR  ->  ( D  +  1 )  e.  RR )
2018, 19syl 17 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( D  +  1 )  e.  RR )
214nnge1d 10652 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <_  D )
2218ltp1d 10537 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  <  ( D  +  1 ) )
2317, 18, 20, 21, 22lelttrd 9793 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  ( D  +  1 ) )
24 sq1 12369 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2524a1i 11 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  =  1 )
264nncnd 10625 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  CC )
27 peano2cn 9805 . . . . . . . 8  |-  ( D  e.  CC  ->  ( D  +  1 )  e.  CC )
2826, 27syl 17 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( D  +  1 )  e.  CC )
2928sqsqrtd 13501 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) ) ^ 2 )  =  ( D  + 
1 ) )
3023, 25, 293brtr4d 4433 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  <  ( ( sqr `  ( D  +  1 ) ) ^ 2 ) )
31 0le1 10137 . . . . . . 7  |-  0  <_  1
3231a1i 11 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  1 )
339rpge0d 11345 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  ( sqr `  ( D  +  1 ) ) )
3417, 10, 32, 33lt2sqd 12450 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1  <  ( sqr `  ( D  +  1 ) )  <->  ( 1 ^ 2 )  < 
( ( sqr `  ( D  +  1 ) ) ^ 2 ) ) )
3530, 34mpbird 236 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  ( sqr `  ( D  +  1 ) ) )
3626sqsqrtd 13501 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  D
) ^ 2 )  =  D )
3721, 25, 363brtr4d 4433 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  <_  ( ( sqr `  D ) ^ 2 ) )
3811rpge0d 11345 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  ( sqr `  D
) )
3917, 12, 32, 38le2sqd 12451 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1  <_  ( sqr `  D )  <->  ( 1 ^ 2 )  <_ 
( ( sqr `  D
) ^ 2 ) ) )
4037, 39mpbird 236 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <_  ( sqr `  D
) )
4117, 17, 10, 12, 35, 40ltleaddd 10234 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1  +  1 )  <  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) ) )
4216, 41syl5eqbr 4436 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  <  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) ) )
43 pell14qrgap 35721 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A )
442, 13, 15, 42, 43ltletrd 9795 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  <  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 985    = wceq 1444    e. wcel 1887    \ cdif 3401   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676   NNcn 10609   2c2 10659   RR+crp 11302   ^cexp 12272   sqrcsqrt 13296  ◻NNcsquarenn 35680  Pell14QRcpell14qr 35683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-pell1qr 35687  df-pell14qr 35688  df-pell1234qr 35689
This theorem is referenced by:  pellfundex  35734
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