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Theorem pellfundval 26965
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 5525 . . . 4  |-  ( a  =  D  ->  (Pell14QR `  a )  =  (Pell14QR `  D ) )
2 rabeq 2782 . . . 4  |-  ( (Pell14QR `  a )  =  (Pell14QR `  D )  ->  { x  e.  (Pell14QR `  a )  |  1  <  x }  =  { x  e.  (Pell14QR `  D )  |  1  <  x } )
31, 2syl 15 . . 3  |-  ( a  =  D  ->  { x  e.  (Pell14QR `  a )  |  1  <  x }  =  { x  e.  (Pell14QR `  D )  |  1  <  x } )
43supeq1d 7199 . 2  |-  ( a  =  D  ->  sup ( { x  e.  (Pell14QR `  a )  |  1  <  x } ,  RR ,  `'  <  )  =  sup ( { x  e.  (Pell14QR `  D
)  |  1  < 
x } ,  RR ,  `'  <  ) )
5 df-pellfund 26930 . 2  |- PellFund  =  ( a  e.  ( NN 
\NN )  |->  sup ( { x  e.  (Pell14QR `  a )  |  1  <  x } ,  RR ,  `'  <  ) )
6 ltso 8903 . . . 4  |-  <  Or  RR
7 cnvso 5214 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
86, 7mpbi 199 . . 3  |-  `'  <  Or  RR
98supex 7214 . 2  |-  sup ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  `'  <  )  e.  _V
104, 5, 9fvmpt 5602 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 1623    e. wcel 1684   {crab 2547    \ cdif 3149   class class class wbr 4023    Or wor 4313   `'ccnv 4688   ` cfv 5255   supcsup 7193   RRcr 8736   1c1 8738    < clt 8867   NNcn 9746  ◻NNcsquarenn 26921  Pell14QRcpell14qr 26924  PellFundcpellfund 26925
This theorem is referenced by:  pellfundre  26966  pellfundge  26967  pellfundlb  26969  pellfundglb  26970
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-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
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-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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-pellfund 26930
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