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Theorem pell1qrgap 26107
Description: First-quadrant Pell solutions are bounded away from 1. (This particular bound allows us to prove exact values for the fundamental solution later.) (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell1qrgap  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A )

Proof of Theorem pell1qrgap
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpell1qr 26080 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1QR `  D )  <->  ( A  e.  RR  /\  E. a  e.  NN0  E. b  e. 
NN0  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
21adantr 451 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  1  <  A
)  ->  ( A  e.  (Pell1QR `  D )  <->  ( A  e.  RR  /\  E. a  e.  NN0  E. b  e.  NN0  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
3 eldifi 3332 . . . . . . . . . . . 12  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
43adantr 451 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  1  <  A
)  ->  D  e.  NN )
54ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  1  <  A )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  NN0 )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  D  e.  NN )
6 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  1  <  A )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  NN0 )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( a  e.  NN0  /\  b  e.  NN0 )
)
7 simpr 447 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  1  <  A
)  ->  1  <  A )
87ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  1  <  A )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  NN0 )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
1  <  A )
9 simprl 732 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  1  <  A )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  NN0 )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  A  =  ( a  +  ( ( sqr `  D )  x.  b
) ) )
108, 9breqtrd 4084 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  1  <  A )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  NN0 )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
1  <  ( a  +  ( ( sqr `  D )  x.  b
) ) )
11 simprr 733 . . . . . . . . . 10  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  1  <  A )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  NN0 )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )
12 pell1qrgaplem 26106 . . . . . . . . . 10  |-  ( ( ( D  e.  NN  /\  ( a  e.  NN0  /\  b  e.  NN0 )
)  /\  ( 1  <  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_ 
( a  +  ( ( sqr `  D
)  x.  b ) ) )
135, 6, 10, 11, 12syl22anc 1183 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  1  <  A )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  NN0 )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_ 
( a  +  ( ( sqr `  D
)  x.  b ) ) )
1413, 9breqtrrd 4086 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  1  <  A )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  NN0 )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A )
1514ex 423 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  1  < 
A )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  NN0 ) )  ->  ( ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) )
1615rexlimdvva 2708 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  1  < 
A )  /\  A  e.  RR )  ->  ( E. a  e.  NN0  E. b  e.  NN0  ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 )  -> 
( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) )
1716expimpd 586 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  1  <  A
)  ->  ( ( A  e.  RR  /\  E. a  e.  NN0  E. b  e.  NN0  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) )
182, 17sylbid 206 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  1  <  A
)  ->  ( A  e.  (Pell1QR `  D )  ->  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) )
1918ex 423 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( 1  <  A  ->  ( A  e.  (Pell1QR `  D )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) ) )
2019com23 72 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1QR `  D )  ->  (
1  <  A  ->  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) ) )
21203imp 1145 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   E.wrex 2578    \ cdif 3183   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   RRcr 8781   1c1 8783    + caddc 8785    x. cmul 8787    < clt 8912    <_ cle 8913    - cmin 9082   NNcn 9791   2c2 9840   NN0cn0 10012   ^cexp 11151   sqrcsqr 11765  ◻NNcsquarenn 26069  Pell1QRcpell1qr 26070
This theorem is referenced by:  pell14qrgap  26108
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
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