Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pell1qrgap Structured version   Unicode version

Theorem pell1qrgap 26939
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 26912 . . . . . 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 453 . . . . 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 3471 . . . . . . . . . . 11  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
43ad4antr 714 . . . . . . . . . 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 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  e.  NN0  /\  b  e.  NN0 )
)
6 simp-4r 745 . . . . . . . . . . 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 734 . . . . . . . . . . 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 4238 . . . . . . . . . 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 735 . . . . . . . . . 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 26938 . . . . . . . . . 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 1186 . . . . . . . . 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 4240 . . . . . . . 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 425 . . . . . . 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 2839 . . . . . 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 588 . . . . 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 208 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  1  <  A
)  ->  ( A  e.  (Pell1QR `  D )  ->  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) )
1716ex 425 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( 1  <  A  ->  ( A  e.  (Pell1QR `  D )  ->  (
( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) ) )
1817com23 75 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1QR `  D )  ->  (
1  <  A  ->  ( ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A ) ) )
19183imp 1148 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708    \ cdif 3319   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   RRcr 8991   1c1 8993    + caddc 8995    x. cmul 8997    < clt 9122    <_ cle 9123    - cmin 9293   NNcn 10002   2c2 10051   NN0cn0 10223   ^cexp 11384   sqrcsqr 12040  ◻NNcsquarenn 26901  Pell1QRcpell1qr 26902
This theorem is referenced by:  pell14qrgap  26940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-pell1qr 26907
  Copyright terms: Public domain W3C validator