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Theorem pellfund14gap 29399
Description: There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pellfund14gap  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  A  =  1 )

Proof of Theorem pellfund14gap
StepHypRef Expression
1 simp3r 1017 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  A  <  (PellFund `  D )
)
2 pell14qrre 29369 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
323adant3 1008 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  A  e.  RR )
4 pellfundre 29393 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
543ad2ant1 1009 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  (PellFund `  D )  e.  RR )
63, 5ltnled 9636 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  ( A  <  (PellFund `  D )  <->  -.  (PellFund `  D )  <_  A ) )
71, 6mpbid 210 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  -.  (PellFund `  D )  <_  A )
8 simpl1 991 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  ( 1  <_  A  /\  A  <  (PellFund `  D
) ) )  /\  1  <  A )  ->  D  e.  ( NN  \NN )
)
9 simpl2 992 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  ( 1  <_  A  /\  A  <  (PellFund `  D
) ) )  /\  1  <  A )  ->  A  e.  (Pell14QR `  D
) )
10 simpr 461 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  ( 1  <_  A  /\  A  <  (PellFund `  D
) ) )  /\  1  <  A )  -> 
1  <  A )
11 pellfundlb 29396 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  <_  A
)
128, 9, 10, 11syl3anc 1219 . . . 4  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  ( 1  <_  A  /\  A  <  (PellFund `  D
) ) )  /\  1  <  A )  -> 
(PellFund `  D )  <_  A )
137, 12mtand 659 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  -.  1  <  A )
14 simp3l 1016 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  1  <_  A )
15 1re 9500 . . . . 5  |-  1  e.  RR
16 leloe 9576 . . . . 5  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  ( 1  <_  A  <->  ( 1  <  A  \/  1  =  A )
) )
1715, 3, 16sylancr 663 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  (
1  <_  A  <->  ( 1  <  A  \/  1  =  A ) ) )
1814, 17mpbid 210 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  (
1  <  A  \/  1  =  A )
)
19 orel1 382 . . 3  |-  ( -.  1  <  A  -> 
( ( 1  < 
A  \/  1  =  A )  ->  1  =  A ) )
2013, 18, 19sylc 60 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  1  =  A )
2120eqcomd 2462 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  A  =  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    \ cdif 3436   class class class wbr 4403   ` cfv 5529   RRcr 9396   1c1 9398    < clt 9533    <_ cle 9534   NNcn 10437  ◻NNcsquarenn 29348  Pell14QRcpell14qr 29351  PellFundcpellfund 29352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-omul 7038  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-oi 7839  df-card 8224  df-acn 8227  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-q 11069  df-rp 11107  df-ico 11421  df-fz 11559  df-fl 11763  df-mod 11830  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-dvds 13658  df-gcd 13813  df-numer 13935  df-denom 13936  df-squarenn 29353  df-pell1qr 29354  df-pell14qr 29355  df-pell1234qr 29356  df-pellfund 29357
This theorem is referenced by:  pellfund14  29410
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