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Theorem pellfundval 30436
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 5865 . . . 4  |-  ( a  =  D  ->  (Pell14QR `  a )  =  (Pell14QR `  D ) )
2 rabeq 3107 . . . 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 7905 . 2  |-  ( a  =  D  ->  sup ( { x  e.  (Pell14QR `  a )  |  1  <  x } ,  RR ,  `'  <  )  =  sup ( { x  e.  (Pell14QR `  D
)  |  1  < 
x } ,  RR ,  `'  <  ) )
5 df-pellfund 30401 . 2  |- PellFund  =  ( a  e.  ( NN 
\NN )  |->  sup ( { x  e.  (Pell14QR `  a )  |  1  <  x } ,  RR ,  `'  <  ) )
6 ltso 9664 . . . 4  |-  <  Or  RR
7 cnvso 5545 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
86, 7mpbi 208 . . 3  |-  `'  <  Or  RR
98supex 7922 . 2  |-  sup ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  `'  <  )  e.  _V
104, 5, 9fvmpt 5949 1  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  =  sup ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {crab 2818    \ cdif 3473   class class class wbr 4447    Or wor 4799   `'ccnv 4998   ` cfv 5587   supcsup 7899   RRcr 9490   1c1 9492    < clt 9627   NNcn 10535  ◻NNcsquarenn 30392  Pell14QRcpell14qr 30395  PellFundcpellfund 30396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-resscn 9548  ax-pre-lttri 9565  ax-pre-lttrn 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7900  df-pnf 9629  df-mnf 9630  df-ltxr 9632  df-pellfund 30401
This theorem is referenced by:  pellfundre  30437  pellfundge  30438  pellfundlb  30440  pellfundglb  30441
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