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Theorem pellfund14gap 31006
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 30976 . . . . . . 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 31000 . . . . . . 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 9749 . . . . 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 31003 . . . . 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 9612 . . . . 5  |-  1  e.  RR
16 leloe 9688 . . . . 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 2465 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 1395    e. wcel 1819    \ cdif 3468   class class class wbr 4456   ` cfv 5594   RRcr 9508   1c1 9510    < clt 9645    <_ cle 9646   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-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
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-reu 2814  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-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  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-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-ico 11560  df-fz 11698  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-numer 14280  df-denom 14281  df-squarenn 30960  df-pell1qr 30961  df-pell14qr 30962  df-pell1234qr 30963  df-pellfund 30964
This theorem is referenced by:  pellfund14  31017
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