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Theorem pellfundlb 27072
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 27068 . . 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 3271 . . . . 5  |-  { a  e.  (Pell14QR `  D
)  |  1  < 
a }  C_  (Pell14QR `  D )
4 pell14qrre 27045 . . . . . . 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 3198 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  RR )
73, 6syl5ss 3203 . . . 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 8853 . . . 4  |-  1  e.  RR
10 breq2 4043 . . . . . . . 8  |-  ( a  =  c  ->  (
1  <  a  <->  1  <  c ) )
1110elrab 2936 . . . . . . 7  |-  ( c  e.  { a  e.  (Pell14QR `  D )  |  1  <  a } 
<->  ( c  e.  (Pell14QR `  D )  /\  1  <  c ) )
12 pell14qrre 27045 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  c  e.  (Pell14QR `  D ) )  -> 
c  e.  RR )
13 ltle 8926 . . . . . . . . 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 2638 . . . . 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 4042 . . . . . 6  |-  ( b  =  1  ->  (
b  <_  c  <->  1  <_  c ) )
2019ralbidv 2576 . . . . 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 2897 . . . 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 4043 . . . . 5  |-  ( a  =  A  ->  (
1  <  a  <->  1  <  A ) )
2625elrab 2936 . . . 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 9751 . . 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 4059 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    C_ wss 3165   class class class wbr 4039   `'ccnv 4704   ` cfv 5271   supcsup 7209   RRcr 8752   1c1 8754    < clt 8883    <_ cle 8884   NNcn 9762  ◻NNcsquarenn 27024  Pell14QRcpell14qr 27027  PellFundcpellfund 27028
This theorem is referenced by:  pellfundglb  27073  pellfund14gap  27075  rmspecfund  27097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-pell14qr 27031  df-pell1234qr 27032  df-pellfund 27033
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