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Theorem pellfundval 27068
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 5541 . . . 4  |-  ( a  =  D  ->  (Pell14QR `  a )  =  (Pell14QR `  D ) )
2 rabeq 2795 . . . 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 7215 . 2  |-  ( a  =  D  ->  sup ( { x  e.  (Pell14QR `  a )  |  1  <  x } ,  RR ,  `'  <  )  =  sup ( { x  e.  (Pell14QR `  D
)  |  1  < 
x } ,  RR ,  `'  <  ) )
5 df-pellfund 27033 . 2  |- PellFund  =  ( a  e.  ( NN 
\NN )  |->  sup ( { x  e.  (Pell14QR `  a )  |  1  <  x } ,  RR ,  `'  <  ) )
6 ltso 8919 . . . 4  |-  <  Or  RR
7 cnvso 5230 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
86, 7mpbi 199 . . 3  |-  `'  <  Or  RR
98supex 7230 . 2  |-  sup ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  `'  <  )  e.  _V
104, 5, 9fvmpt 5618 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 1632    e. wcel 1696   {crab 2560    \ cdif 3162   class class class wbr 4039    Or wor 4329   `'ccnv 4704   ` cfv 5271   supcsup 7209   RRcr 8752   1c1 8754    < clt 8883   NNcn 9762  ◻NNcsquarenn 27024  Pell14QRcpell14qr 27027  PellFundcpellfund 27028
This theorem is referenced by:  pellfundre  27069  pellfundge  27070  pellfundlb  27072  pellfundglb  27073
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-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
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-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-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-pellfund 27033
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