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Theorem pell14qrgapw 26109
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 9860 . . 3  |-  2  e.  RR
21a1i 10 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  e.  RR )
3 eldifi 3332 . . . . . . . 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 10436 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  RR+ )
6 1rp 10405 . . . . . . 7  |-  1  e.  RR+
76a1i 10 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  e.  RR+ )
85, 7rpaddcld 10452 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( D  +  1 )  e.  RR+ )
98rpsqrcld 11941 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  ( D  + 
1 ) )  e.  RR+ )
109rpred 10437 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  ( D  + 
1 ) )  e.  RR )
115rpsqrcld 11941 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  D )  e.  RR+ )
1211rpred 10437 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  ( sqr `  D )  e.  RR )
1310, 12readdcld 8907 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  e.  RR )
14 pell14qrre 26090 . . 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 9849 . . 3  |-  2  =  ( 1  +  1 )
17 1re 8882 . . . . 5  |-  1  e.  RR
1817a1i 10 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  e.  RR )
194nnred 9806 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  RR )
20 peano2re 9030 . . . . . . . 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 9833 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <_  D )
2319ltp1d 9732 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  <  ( D  +  1 ) )
2418, 19, 21, 22, 23lelttrd 9019 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  ( D  +  1 ) )
25 sq1 11245 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2625a1i 10 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  =  1 )
274nncnd 9807 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  D  e.  CC )
28 peano2cn 9029 . . . . . . . 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 11968 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) ) ^ 2 )  =  ( D  + 
1 ) )
3124, 26, 303brtr4d 4090 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  <  ( ( sqr `  ( D  +  1 ) ) ^ 2 ) )
32 0le1 9342 . . . . . . 7  |-  0  <_  1
3332a1i 10 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  1 )
349rpge0d 10441 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  ( sqr `  ( D  +  1 ) ) )
3518, 10, 33, 34lt2sqd 11326 . . . . 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 11968 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  D
) ^ 2 )  =  D )
3822, 26, 373brtr4d 4090 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1 ^ 2 )  <_  ( ( sqr `  D ) ^ 2 ) )
3911rpge0d 10441 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  0  <_  ( sqr `  D
) )
4018, 12, 33, 39le2sqd 11327 . . . . 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 9437 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
1  +  1 )  <  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) ) )
4316, 42syl5eqbr 4093 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  <  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) ) )
44 pell14qrgap 26108 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A )
452, 13, 15, 43, 44ltletrd 9021 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 1633    e. wcel 1701    \ cdif 3183   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    < clt 8912    <_ cle 8913   NNcn 9791   2c2 9840   RR+crp 10401   ^cexp 11151   sqrcsqr 11765  ◻NNcsquarenn 26069  Pell14QRcpell14qr 26072
This theorem is referenced by:  pellfundex  26119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-seq 11094  df-exp 11152  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-pell1qr 26075  df-pell14qr 26076  df-pell1234qr 26077
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