Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pellfundval Structured version   Unicode version

Theorem pellfundval 30999
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 5872 . . . 4  |-  ( a  =  D  ->  (Pell14QR `  a )  =  (Pell14QR `  D ) )
2 rabeq 3103 . . . 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 7923 . 2  |-  ( a  =  D  ->  sup ( { x  e.  (Pell14QR `  a )  |  1  <  x } ,  RR ,  `'  <  )  =  sup ( { x  e.  (Pell14QR `  D
)  |  1  < 
x } ,  RR ,  `'  <  ) )
5 df-pellfund 30964 . 2  |- PellFund  =  ( a  e.  ( NN 
\NN )  |->  sup ( { x  e.  (Pell14QR `  a )  |  1  <  x } ,  RR ,  `'  <  ) )
6 gtso 9683 . . 3  |-  `'  <  Or  RR
76supex 7940 . 2  |-  sup ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  `'  <  )  e.  _V
84, 5, 7fvmpt 5956 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 1395    e. wcel 1819   {crab 2811    \ cdif 3468   class class class wbr 4456   `'ccnv 5007   ` cfv 5594   supcsup 7918   RRcr 9508   1c1 9510    < clt 9645   NNcn 10556  ◻NNcsquarenn 30955  Pell14QRcpell14qr 30958  PellFundcpellfund 30959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-pellfund 30964
This theorem is referenced by:  pellfundre  31000  pellfundge  31001  pellfundlb  31003  pellfundglb  31004
  Copyright terms: Public domain W3C validator