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Theorem pellfundval 26943
Description: Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pellfundval  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  `'  <  ) )
Distinct variable group:    x, D

Proof of Theorem pellfundval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . . . 4  |-  ( a  =  D  ->  (Pell14QR `  a )  =  (Pell14QR `  D ) )
2 rabeq 2950 . . . 4  |-  ( (Pell14QR `  a )  =  (Pell14QR `  D )  ->  { x  e.  (Pell14QR `  a )  |  1  <  x }  =  { x  e.  (Pell14QR `  D )  |  1  <  x } )
31, 2syl 16 . . 3  |-  ( a  =  D  ->  { x  e.  (Pell14QR `  a )  |  1  <  x }  =  { x  e.  (Pell14QR `  D )  |  1  <  x } )
43supeq1d 7451 . 2  |-  ( a  =  D  ->  sup ( { x  e.  (Pell14QR `  a )  |  1  <  x } ,  RR ,  `'  <  )  =  sup ( { x  e.  (Pell14QR `  D
)  |  1  < 
x } ,  RR ,  `'  <  ) )
5 df-pellfund 26908 . 2  |- PellFund  =  ( a  e.  ( NN 
\NN )  |->  sup ( { x  e.  (Pell14QR `  a )  |  1  <  x } ,  RR ,  `'  <  ) )
6 ltso 9156 . . . 4  |-  <  Or  RR
7 cnvso 5411 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
86, 7mpbi 200 . . 3  |-  `'  <  Or  RR
98supex 7468 . 2  |-  sup ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  `'  <  )  e.  _V
104, 5, 9fvmpt 5806 1  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2709    \ cdif 3317   class class class wbr 4212    Or wor 4502   `'ccnv 4877   ` cfv 5454   supcsup 7445   RRcr 8989   1c1 8991    < clt 9120   NNcn 10000  ◻NNcsquarenn 26899  Pell14QRcpell14qr 26902  PellFundcpellfund 26903
This theorem is referenced by:  pellfundre  26944  pellfundge  26945  pellfundlb  26947  pellfundglb  26948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-pellfund 26908
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