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Theorem pellfund14gap 30751
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 1025 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  A  <  (PellFund `  D )
)
2 pell14qrre 30721 . . . . . . 7  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
323adant3 1016 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  A  e.  RR )
4 pellfundre 30745 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
(PellFund `  D )  e.  RR )
543ad2ant1 1017 . . . . . 6  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  (PellFund `  D )  e.  RR )
63, 5ltnled 9743 . . . . 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 999 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  ( 1  <_  A  /\  A  <  (PellFund `  D
) ) )  /\  1  <  A )  ->  D  e.  ( NN  \NN )
)
9 simpl2 1000 . . . . 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 30748 . . . . 5  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  <_  A
)
128, 9, 10, 11syl3anc 1228 . . . 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 1024 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D )  /\  (
1  <_  A  /\  A  <  (PellFund `  D )
) )  ->  1  <_  A )
15 1re 9607 . . . . 5  |-  1  e.  RR
16 leloe 9683 . . . . 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 2475 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 973    = wceq 1379    e. wcel 1767    \ cdif 3478   class class class wbr 4453   ` cfv 5594   RRcr 9503   1c1 9505    < clt 9640    <_ cle 9641   NNcn 10548  ◻NNcsquarenn 30700  Pell14QRcpell14qr 30703  PellFundcpellfund 30704
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-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-omul 7147  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-ico 11547  df-fz 11685  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-numer 14144  df-denom 14145  df-squarenn 30705  df-pell1qr 30706  df-pell14qr 30707  df-pell1234qr 30708  df-pellfund 30709
This theorem is referenced by:  pellfund14  30762
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