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Theorem pellfundglb 26939
 Description: If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pellfundglb NN PellFund Pell1QRPellFund
Distinct variable groups:   ,   ,

Proof of Theorem pellfundglb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pellfundval 26934 . . . . . . 7 NN PellFund Pell14QR
213ad2ant1 978 . . . . . 6 NN PellFund PellFund Pell14QR
3 simp3 959 . . . . . 6 NN PellFund PellFund
42, 3eqbrtrrd 4226 . . . . 5 NN PellFund Pell14QR
5 pellfundre 26935 . . . . . . . 8 NN PellFund
653ad2ant1 978 . . . . . . 7 NN PellFund PellFund
72, 6eqeltrrd 2510 . . . . . 6 NN PellFund Pell14QR
8 simp2 958 . . . . . 6 NN PellFund
97, 8ltnled 9212 . . . . 5 NN PellFund Pell14QR Pell14QR
104, 9mpbid 202 . . . 4 NN PellFund Pell14QR
11 ssrab2 3420 . . . . . 6 Pell14QR Pell14QR
12 pell14qrre 26911 . . . . . . . . 9 NN Pell14QR
1312ex 424 . . . . . . . 8 NN Pell14QR
1413ssrdv 3346 . . . . . . 7 NN Pell14QR
15143ad2ant1 978 . . . . . 6 NN PellFund Pell14QR
1611, 15syl5ss 3351 . . . . 5 NN PellFund Pell14QR
17 pell1qrss14 26922 . . . . . . . 8 NN Pell1QR Pell14QR
18173ad2ant1 978 . . . . . . 7 NN PellFund Pell1QR Pell14QR
19 pellqrex 26933 . . . . . . . 8 NN Pell1QR
20193ad2ant1 978 . . . . . . 7 NN PellFund Pell1QR
21 ssrexv 3400 . . . . . . 7 Pell1QR Pell14QR Pell1QR Pell14QR
2218, 20, 21sylc 58 . . . . . 6 NN PellFund Pell14QR
23 rabn0 3639 . . . . . 6 Pell14QR Pell14QR
2422, 23sylibr 204 . . . . 5 NN PellFund Pell14QR
25 infmrgelbi 26932 . . . . . 6 Pell14QR Pell14QR Pell14QR Pell14QR
2625ex 424 . . . . 5 Pell14QR Pell14QR Pell14QR Pell14QR
2716, 24, 8, 26syl3anc 1184 . . . 4 NN PellFund Pell14QR Pell14QR
2810, 27mtod 170 . . 3 NN PellFund Pell14QR
29 rexnal 2708 . . 3 Pell14QR Pell14QR
3028, 29sylibr 204 . 2 NN PellFund Pell14QR
31 breq2 4208 . . . . . . . 8
3231elrab 3084 . . . . . . 7 Pell14QR Pell14QR
33 simprl 733 . . . . . . . . 9 NN PellFund Pell14QR Pell14QR
34 1re 9082 . . . . . . . . . . 11
3534a1i 11 . . . . . . . . . 10 NN PellFund Pell14QR
36 simpl1 960 . . . . . . . . . . 11 NN PellFund Pell14QR NN
37 pell14qrre 26911 . . . . . . . . . . 11 NN Pell14QR
3836, 33, 37syl2anc 643 . . . . . . . . . 10 NN PellFund Pell14QR
39 simprr 734 . . . . . . . . . 10 NN PellFund Pell14QR
4035, 38, 39ltled 9213 . . . . . . . . 9 NN PellFund Pell14QR
4133, 40jca 519 . . . . . . . 8 NN PellFund Pell14QR Pell14QR
42 elpell1qr2 26926 . . . . . . . . 9 NN Pell1QR Pell14QR
4336, 42syl 16 . . . . . . . 8 NN PellFund Pell14QR Pell1QR Pell14QR
4441, 43mpbird 224 . . . . . . 7 NN PellFund Pell14QR Pell1QR
4532, 44sylan2b 462 . . . . . 6 NN PellFund Pell14QR Pell1QR
4645adantrr 698 . . . . 5 NN PellFund Pell14QR Pell1QR
47 simpl1 960 . . . . . . 7 NN PellFund Pell14QR NN
48 simprl 733 . . . . . . . 8 NN PellFund Pell14QR Pell14QR
4911, 48sseldi 3338 . . . . . . 7 NN PellFund Pell14QR Pell14QR
50 simpr 448 . . . . . . . . . . 11 Pell14QR
5150a1i 11 . . . . . . . . . 10 NN PellFund Pell14QR
5232, 51syl5bi 209 . . . . . . . . 9 NN PellFund Pell14QR
5352imp 419 . . . . . . . 8 NN PellFund Pell14QR
5453adantrr 698 . . . . . . 7 NN PellFund Pell14QR
55 pellfundlb 26938 . . . . . . 7 NN Pell14QR PellFund
5647, 49, 54, 55syl3anc 1184 . . . . . 6 NN PellFund Pell14QR PellFund
57 simprr 734 . . . . . . 7 NN PellFund Pell14QR
5815adantr 452 . . . . . . . . 9 NN PellFund Pell14QR Pell14QR
5958, 49sseldd 3341 . . . . . . . 8 NN PellFund Pell14QR
60 simpl2 961 . . . . . . . 8 NN PellFund Pell14QR
6159, 60ltnled 9212 . . . . . . 7 NN PellFund Pell14QR
6257, 61mpbird 224 . . . . . 6 NN PellFund Pell14QR
6356, 62jca 519 . . . . 5 NN PellFund Pell14QR PellFund
6446, 63jca 519 . . . 4 NN PellFund Pell14QR Pell1QR PellFund
6564ex 424 . . 3 NN PellFund Pell14QR Pell1QR PellFund
6665reximdv2 2807 . 2 NN PellFund Pell14QR Pell1QRPellFund
6730, 66mpd 15 1 NN PellFund Pell1QRPellFund
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2598  wral 2697  wrex 2698  crab 2701   cdif 3309   wss 3312  c0 3620   class class class wbr 4204  ccnv 4869  cfv 5446  csup 7437  cr 8981  c1 8983   clt 9112   cle 9113  cn 9992  ◻NNcsquarenn 26890  Pell1QRcpell1qr 26891  Pell14QRcpell14qr 26893  PellFundcpellfund 26894 This theorem is referenced by:  pellfundex  26940 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-acn 7821  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-ico 10914  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-numer 13119  df-denom 13120  df-squarenn 26895  df-pell1qr 26896  df-pell14qr 26897  df-pell1234qr 26898  df-pellfund 26899
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