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Theorem pellfundlb 26638
Description: A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
Assertion
Ref Expression
pellfundlb  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  <_  A
)

Proof of Theorem pellfundlb
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pellfundval 26634 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
213ad2ant1 978 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) )
3 ssrab2 3371 . . . . 5  |-  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  (Pell14QR `  D )
4 pell14qrre 26611 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  d  e.  (Pell14QR `  D ) )  -> 
d  e.  RR )
54ex 424 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( d  e.  (Pell14QR `  D )  ->  d  e.  RR ) )
65ssrdv 3297 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  RR )
73, 6syl5ss 3302 . . . 4  |-  ( D  e.  ( NN  \NN )  ->  { a  e.  (Pell14QR `  D )  |  1  <  a }  C_  RR )
873ad2ant1 978 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  RR )
9 1re 9023 . . . 4  |-  1  e.  RR
10 breq2 4157 . . . . . . . 8  |-  ( a  =  c  ->  (
1  <  a  <->  1  <  c ) )
1110elrab 3035 . . . . . . 7  |-  ( c  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( c  e.  (Pell14QR `  D )  /\  1  <  c ) )
12 pell14qrre 26611 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  c  e.  (Pell14QR `  D ) )  -> 
c  e.  RR )
13 ltle 9096 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  c  e.  RR )  ->  ( 1  <  c  ->  1  <_  c )
)
149, 12, 13sylancr 645 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  c  e.  (Pell14QR `  D ) )  -> 
( 1  <  c  ->  1  <_  c )
)
1514expimpd 587 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
( ( c  e.  (Pell14QR `  D )  /\  1  <  c )  ->  1  <_  c
) )
1611, 15syl5bi 209 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( c  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  ->  1  <_  c ) )
1716ralrimiv 2731 . . . . 5  |-  ( D  e.  ( NN  \NN )  ->  A. c  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } 1  <_ 
c )
18173ad2ant1 978 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A. c  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 1  <_  c )
19 breq1 4156 . . . . . 6  |-  ( b  =  1  ->  (
b  <_  c  <->  1  <_  c ) )
2019ralbidv 2669 . . . . 5  |-  ( b  =  1  ->  ( A. c  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } b  <_ 
c  <->  A. c  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a } 1  <_ 
c ) )
2120rspcev 2995 . . . 4  |-  ( ( 1  e.  RR  /\  A. c  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } 1  <_ 
c )  ->  E. b  e.  RR  A. c  e. 
{ a  e.  (Pell14QR `  D )  |  1  <  a } b  <_  c )
229, 18, 21sylancr 645 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  E. b  e.  RR  A. c  e. 
{ a  e.  (Pell14QR `  D )  |  1  <  a } b  <_  c )
23 simp2 958 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  (Pell14QR `  D )
)
24 simp3 959 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  A )
25 breq2 4157 . . . . 5  |-  ( a  =  A  ->  (
1  <  a  <->  1  <  A ) )
2625elrab 3035 . . . 4  |-  ( A  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( A  e.  (Pell14QR `  D )  /\  1  <  A ) )
2723, 24, 26sylanbrc 646 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } )
28 infmrlb 9921 . . 3  |-  ( ( { a  e.  (Pell14QR `  D )  |  1  <  a }  C_  RR  /\  E. b  e.  RR  A. c  e. 
{ a  e.  (Pell14QR `  D )  |  1  <  a } b  <_  c  /\  A  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } )  ->  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  )  <_  A )
298, 22, 27, 28syl3anc 1184 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  <_  A )
302, 29eqbrtrd 4173 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  <_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650   {crab 2653    \ cdif 3260    C_ wss 3263   class class class wbr 4153   `'ccnv 4817   ` cfv 5394   supcsup 7380   RRcr 8922   1c1 8924    < clt 9053    <_ cle 9054   NNcn 9932  ◻NNcsquarenn 26590  Pell14QRcpell14qr 26593  PellFundcpellfund 26594
This theorem is referenced by:  pellfundglb  26639  pellfund14gap  26641  rmspecfund  26663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-n0 10154  df-z 10215  df-pell14qr 26597  df-pell1234qr 26598  df-pellfund 26599
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