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Theorem pell1qrgap 35639
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 35612 . . . . . 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 466 . . . . 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 3587 . . . . . . . . . . 11  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
43ad4antr 736 . . . . . . . . . 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 )
5 simplr 760 . . . . . . . . . 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 )
)
6 simp-4r 775 . . . . . . . . . . 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 )
7 simprl 762 . . . . . . . . . . 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
) ) )
86, 7breqtrd 4445 . . . . . . . . . 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
) ) )
9 simprr 764 . . . . . . . . . 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 )
10 pell1qrgaplem 35638 . . . . . . . . . 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 ) ) )
114, 5, 8, 9, 10syl22anc 1265 . . . . . . . . 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 ) ) )
1211, 7breqtrrd 4447 . . . . . . . 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 )
1312ex 435 . . . . . . 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 ) )
1413rexlimdvva 2924 . . . . . 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 ) )
1514expimpd 606 . . . . 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 ) )
162, 15sylbid 218 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  1  <  A
)  ->  ( A  e.  (Pell1QR `  D )  ->  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) )
1716ex 435 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( 1  <  A  ->  ( A  e.  (Pell1QR `  D )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) ) )
1817com23 81 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1QR `  D )  ->  (
1  <  A  ->  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) ) )
19183imp 1199 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1QR `  D )  /\  1  <  A )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   E.wrex 2776    \ cdif 3433   class class class wbr 4420   ` cfv 5597  (class class class)co 6301   RRcr 9538   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860   NNcn 10609   2c2 10659   NN0cn0 10869   ^cexp 12271   sqrcsqrt 13284  ◻NNcsquarenn 35599  Pell1QRcpell1qr 35600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  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 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-sup 7958  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 12213  df-exp 12272  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-pell1qr 35606
This theorem is referenced by:  pell14qrgap  35640
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