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Theorem pellfundlb 26969
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 26965 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D
)  |  1  < 
a } ,  RR ,  `'  <  ) )
213ad2ant1 976 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  =  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  ) )
3 ssrab2 3258 . . . . 5  |-  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  (Pell14QR `  D )
4 pell14qrre 26942 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  d  e.  (Pell14QR `  D ) )  -> 
d  e.  RR )
54ex 423 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( d  e.  (Pell14QR `  D )  ->  d  e.  RR ) )
65ssrdv 3185 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  RR )
73, 6syl5ss 3190 . . . 4  |-  ( D  e.  ( NN  \NN )  ->  { a  e.  (Pell14QR `  D )  |  1  <  a }  C_  RR )
873ad2ant1 976 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  RR )
9 1re 8837 . . . 4  |-  1  e.  RR
10 breq2 4027 . . . . . . . 8  |-  ( a  =  c  ->  (
1  <  a  <->  1  <  c ) )
1110elrab 2923 . . . . . . 7  |-  ( c  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( c  e.  (Pell14QR `  D )  /\  1  <  c ) )
12 pell14qrre 26942 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  c  e.  (Pell14QR `  D ) )  -> 
c  e.  RR )
13 ltle 8910 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  c  e.  RR )  ->  ( 1  <  c  ->  1  <_  c )
)
149, 12, 13sylancr 644 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  c  e.  (Pell14QR `  D ) )  -> 
( 1  <  c  ->  1  <_  c )
)
1514expimpd 586 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
( ( c  e.  (Pell14QR `  D )  /\  1  <  c )  ->  1  <_  c
) )
1611, 15syl5bi 208 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( c  e.  {
a  e.  (Pell14QR `  D
)  |  1  < 
a }  ->  1  <_  c ) )
1716ralrimiv 2625 . . . . 5  |-  ( D  e.  ( NN  \NN )  ->  A. c  e.  { a  e.  (Pell14QR `  D
)  |  1  < 
a } 1  <_ 
c )
18173ad2ant1 976 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A. c  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 1  <_  c )
19 breq1 4026 . . . . . 6  |-  ( b  =  1  ->  (
b  <_  c  <->  1  <_  c ) )
2019ralbidv 2563 . . . . 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 2884 . . . 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 644 . . 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 956 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  (Pell14QR `  D )
)
24 simp3 957 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  1  <  A )
25 breq2 4027 . . . . 5  |-  ( a  =  A  ->  (
1  <  a  <->  1  <  A ) )
2625elrab 2923 . . . 4  |-  ( A  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( A  e.  (Pell14QR `  D )  /\  1  <  A ) )
2723, 24, 26sylanbrc 645 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  A  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } )
28 infmrlb 9735 . . 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 1182 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  sup ( { a  e.  (Pell14QR `  D )  |  1  <  a } ,  RR ,  `'  <  )  <_  A )
302, 29eqbrtrd 4043 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149    C_ wss 3152   class class class wbr 4023   `'ccnv 4688   ` cfv 5255   supcsup 7193   RRcr 8736   1c1 8738    < clt 8867    <_ cle 8868   NNcn 9746  ◻NNcsquarenn 26921  Pell14QRcpell14qr 26924  PellFundcpellfund 26925
This theorem is referenced by:  pellfundglb  26970  pellfund14gap  26972  rmspecfund  26994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-pell14qr 26928  df-pell1234qr 26929  df-pellfund 26930
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