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Theorem pellfundlb 31059
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 31055 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
213ad2ant1 1015 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) )
3 ssrab2 3571 . . . . 5  |-  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  (Pell14QR `  D )
4 pell14qrre 31032 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  d  e.  (Pell14QR `  D ) )  -> 
d  e.  RR )
54ex 432 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( d  e.  (Pell14QR `  D )  ->  d  e.  RR ) )
65ssrdv 3495 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  RR )
73, 6syl5ss 3500 . . . 4  |-  ( D  e.  ( NN  \NN )  ->  { a  e.  (Pell14QR `  D )  |  1  <  a }  C_  RR )
873ad2ant1 1015 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  RR )
9 1re 9584 . . . 4  |-  1  e.  RR
10 breq2 4443 . . . . . . . 8  |-  ( a  =  c  ->  (
1  <  a  <->  1  <  c ) )
1110elrab 3254 . . . . . . 7  |-  ( c  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( c  e.  (Pell14QR `  D )  /\  1  <  c ) )
12 pell14qrre 31032 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  c  e.  (Pell14QR `  D ) )  -> 
c  e.  RR )
13 ltle 9662 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  c  e.  RR )  ->  ( 1  <  c  ->  1  <_  c )
)
149, 12, 13sylancr 661 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  c  e.  (Pell14QR `  D ) )  -> 
( 1  <  c  ->  1  <_  c )
)
1514expimpd 601 . . . . . . 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 2866 . . . . 5  |-  ( D  e.  ( NN  \NN )  ->  A. c  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } 1  <_ 
c )
18173ad2ant1 1015 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A. c  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 1  <_  c )
19 breq1 4442 . . . . . 6  |-  ( b  =  1  ->  (
b  <_  c  <->  1  <_  c ) )
2019ralbidv 2893 . . . . 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 3207 . . . 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 661 . . 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 995 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  (Pell14QR `  D )
)
24 simp3 996 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  A )
25 breq2 4443 . . . . 5  |-  ( a  =  A  ->  (
1  <  a  <->  1  <  A ) )
2625elrab 3254 . . . 4  |-  ( A  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( A  e.  (Pell14QR `  D )  /\  1  <  A ) )
2723, 24, 26sylanbrc 662 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } )
28 infmrlb 10519 . . 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 1226 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  <_  A )
302, 29eqbrtrd 4459 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808    \ cdif 3458    C_ wss 3461   class class class wbr 4439   `'ccnv 4987   ` cfv 5570   supcsup 7892   RRcr 9480   1c1 9482    < clt 9617    <_ cle 9618   NNcn 10531  ◻NNcsquarenn 31011  Pell14QRcpell14qr 31014  PellFundcpellfund 31015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-pell14qr 31018  df-pell1234qr 31019  df-pellfund 31020
This theorem is referenced by:  pellfundglb  31060  pellfund14gap  31062  rmspecfund  31084
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