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Theorem pellfundlb 29366
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 29362 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
213ad2ant1 1009 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) )
3 ssrab2 3538 . . . . 5  |-  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  (Pell14QR `  D )
4 pell14qrre 29339 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  d  e.  (Pell14QR `  D ) )  -> 
d  e.  RR )
54ex 434 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( d  e.  (Pell14QR `  D )  ->  d  e.  RR ) )
65ssrdv 3463 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  RR )
73, 6syl5ss 3468 . . . 4  |-  ( D  e.  ( NN  \NN )  ->  { a  e.  (Pell14QR `  D )  |  1  <  a }  C_  RR )
873ad2ant1 1009 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  RR )
9 1re 9489 . . . 4  |-  1  e.  RR
10 breq2 4397 . . . . . . . 8  |-  ( a  =  c  ->  (
1  <  a  <->  1  <  c ) )
1110elrab 3217 . . . . . . 7  |-  ( c  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( c  e.  (Pell14QR `  D )  /\  1  <  c ) )
12 pell14qrre 29339 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  c  e.  (Pell14QR `  D ) )  -> 
c  e.  RR )
13 ltle 9567 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  c  e.  RR )  ->  ( 1  <  c  ->  1  <_  c )
)
149, 12, 13sylancr 663 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  c  e.  (Pell14QR `  D ) )  -> 
( 1  <  c  ->  1  <_  c )
)
1514expimpd 603 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
( ( c  e.  (Pell14QR `  D )  /\  1  <  c )  ->  1  <_  c
) )
1611, 15syl5bi 217 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( c  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  ->  1  <_  c ) )
1716ralrimiv 2823 . . . . 5  |-  ( D  e.  ( NN  \NN )  ->  A. c  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } 1  <_ 
c )
18173ad2ant1 1009 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A. c  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 1  <_  c )
19 breq1 4396 . . . . . 6  |-  ( b  =  1  ->  (
b  <_  c  <->  1  <_  c ) )
2019ralbidv 2841 . . . . 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 3172 . . . 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 663 . . 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 989 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  (Pell14QR `  D )
)
24 simp3 990 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  A )
25 breq2 4397 . . . . 5  |-  ( a  =  A  ->  (
1  <  a  <->  1  <  A ) )
2625elrab 3217 . . . 4  |-  ( A  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( A  e.  (Pell14QR `  D )  /\  1  <  A ) )
2723, 24, 26sylanbrc 664 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } )
28 infmrlb 10415 . . 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 1219 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  <_  A )
302, 29eqbrtrd 4413 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  <_  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   {crab 2799    \ cdif 3426    C_ wss 3429   class class class wbr 4393   `'ccnv 4940   ` cfv 5519   supcsup 7794   RRcr 9385   1c1 9387    < clt 9522    <_ cle 9523   NNcn 10426  ◻NNcsquarenn 29318  Pell14QRcpell14qr 29321  PellFundcpellfund 29322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-pell14qr 29325  df-pell1234qr 29326  df-pellfund 29327
This theorem is referenced by:  pellfundglb  29367  pellfund14gap  29369  rmspecfund  29391
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