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Theorem pellfundex 35805
 Description: The fundamental solution as an infimum is itself a solution, showing that the solution set is discrete. Since the fundamental solution is an infimum, there must be an element ge to Fund and lt 2*Fund. If this element is equal to the fundamental solution we're done, otherwise use the infimum again to find another element which must be ge Fund and lt the first element; their ratio is a group element in (1,2), contradicting pell14qrgapw 35793. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pellfundex NN PellFund Pell1QR

Proof of Theorem pellfundex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2re 10701 . . . 4
2 pellfundre 35800 . . . 4 NN PellFund
3 remulcl 9642 . . . 4 PellFund PellFund
41, 2, 3sylancr 676 . . 3 NN PellFund
5 0red 9662 . . . . . . 7 NN
6 1red 9676 . . . . . . 7 NN
7 0lt1 10157 . . . . . . . 8
87a1i 11 . . . . . . 7 NN
9 pellfundgt1 35802 . . . . . . 7 NN PellFund
105, 6, 2, 8, 9lttrd 9813 . . . . . 6 NN PellFund
112, 10elrpd 11361 . . . . 5 NN PellFund
122, 11ltaddrpd 11394 . . . 4 NN PellFund PellFund PellFund
132recnd 9687 . . . . 5 NN PellFund
14132timesd 10878 . . . 4 NN PellFund PellFund PellFund
1512, 14breqtrrd 4422 . . 3 NN PellFund PellFund
16 pellfundglb 35804 . . 3 NN PellFund PellFund PellFund Pell1QRPellFund PellFund
174, 15, 16mpd3an23 1392 . 2 NN Pell1QRPellFund PellFund
182adantr 472 . . . . . 6 NN Pell1QR PellFund
19 pell1qrss14 35785 . . . . . . . 8 NN Pell1QR Pell14QR
2019sselda 3418 . . . . . . 7 NN Pell1QR Pell14QR
21 pell14qrre 35774 . . . . . . 7 NN Pell14QR
2220, 21syldan 478 . . . . . 6 NN Pell1QR
2318, 22leloed 9795 . . . . 5 NN Pell1QR PellFund PellFund PellFund
24 simp-4l 784 . . . . . . . . 9 NN Pell1QR PellFund PellFund Pell1QR PellFund NN
25 simp-4r 785 . . . . . . . . 9 NN Pell1QR PellFund PellFund Pell1QR PellFund Pell1QR
26 simplr 770 . . . . . . . . 9 NN Pell1QR PellFund PellFund Pell1QR PellFund Pell1QR
27 simprr 774 . . . . . . . . 9 NN Pell1QR PellFund PellFund Pell1QR PellFund
2822ad3antrrr 744 . . . . . . . . . 10 NN Pell1QR PellFund PellFund Pell1QR PellFund
294ad4antr 746 . . . . . . . . . 10 NN Pell1QR PellFund PellFund Pell1QR PellFund PellFund
3019ad4antr 746 . . . . . . . . . . . . 13 NN Pell1QR PellFund PellFund Pell1QR PellFund Pell1QR Pell14QR
3130, 26sseldd 3419 . . . . . . . . . . . 12 NN Pell1QR PellFund PellFund Pell1QR PellFund Pell14QR
32 pell14qrre 35774 . . . . . . . . . . . 12 NN Pell14QR
3324, 31, 32syl2anc 673 . . . . . . . . . . 11 NN Pell1QR PellFund PellFund Pell1QR PellFund
34 remulcl 9642 . . . . . . . . . . 11
351, 33, 34sylancr 676 . . . . . . . . . 10 NN Pell1QR PellFund PellFund Pell1QR PellFund
36 simprr 774 . . . . . . . . . . 11 NN Pell1QR PellFund PellFund PellFund
3736ad2antrr 740 . . . . . . . . . 10 NN Pell1QR PellFund PellFund Pell1QR PellFund PellFund
38 simprl 772 . . . . . . . . . . 11 NN Pell1QR PellFund PellFund Pell1QR PellFund PellFund
392ad4antr 746 . . . . . . . . . . . 12 NN Pell1QR PellFund PellFund Pell1QR PellFund PellFund
401a1i 11 . . . . . . . . . . . 12 NN Pell1QR PellFund PellFund Pell1QR PellFund
41 2pos 10723 . . . . . . . . . . . . 13
4241a1i 11 . . . . . . . . . . . 12 NN Pell1QR PellFund PellFund Pell1QR PellFund
43 lemul2 10480 . . . . . . . . . . . 12 PellFund PellFund PellFund
4439, 33, 40, 42, 43syl112anc 1296 . . . . . . . . . . 11 NN Pell1QR PellFund PellFund Pell1QR PellFund PellFund PellFund
4538, 44mpbid 215 . . . . . . . . . 10 NN Pell1QR PellFund PellFund Pell1QR PellFund PellFund
4628, 29, 35, 37, 45ltletrd 9812 . . . . . . . . 9 NN Pell1QR PellFund PellFund Pell1QR PellFund
47 simp1 1030 . . . . . . . . . 10 NN Pell1QR Pell1QR NN
48193ad2ant1 1051 . . . . . . . . . . . 12 NN Pell1QR Pell1QR Pell1QR Pell14QR
49 simp2l 1056 . . . . . . . . . . . 12 NN Pell1QR Pell1QR Pell1QR
5048, 49sseldd 3419 . . . . . . . . . . 11 NN Pell1QR Pell1QR Pell14QR
51 simp2r 1057 . . . . . . . . . . . 12 NN Pell1QR Pell1QR Pell1QR
5248, 51sseldd 3419 . . . . . . . . . . 11 NN Pell1QR Pell1QR Pell14QR
53 pell14qrdivcl 35782 . . . . . . . . . . 11 NN Pell14QR Pell14QR Pell14QR
5447, 50, 52, 53syl3anc 1292 . . . . . . . . . 10 NN Pell1QR Pell1QR Pell14QR
5547, 52, 32syl2anc 673 . . . . . . . . . . . . . 14 NN Pell1QR Pell1QR
5655recnd 9687 . . . . . . . . . . . . 13 NN Pell1QR Pell1QR
5756mulid2d 9679 . . . . . . . . . . . 12 NN Pell1QR Pell1QR
58 simp3l 1058 . . . . . . . . . . . 12 NN Pell1QR Pell1QR
5957, 58eqbrtrd 4416 . . . . . . . . . . 11 NN Pell1QR Pell1QR
60 1red 9676 . . . . . . . . . . . 12 NN Pell1QR Pell1QR
6147, 50, 21syl2anc 673 . . . . . . . . . . . 12 NN Pell1QR Pell1QR
62 pell14qrgt0 35776 . . . . . . . . . . . . 13 NN Pell14QR
6347, 52, 62syl2anc 673 . . . . . . . . . . . 12 NN Pell1QR Pell1QR
64 ltmuldiv 10500 . . . . . . . . . . . 12
6560, 61, 55, 63, 64syl112anc 1296 . . . . . . . . . . 11 NN Pell1QR Pell1QR
6659, 65mpbid 215 . . . . . . . . . 10 NN Pell1QR Pell1QR
67 simp3r 1059 . . . . . . . . . . 11 NN Pell1QR Pell1QR
681a1i 11 . . . . . . . . . . . 12 NN Pell1QR Pell1QR
69 ltdivmul2 10504 . . . . . . . . . . . 12
7061, 68, 55, 63, 69syl112anc 1296 . . . . . . . . . . 11 NN Pell1QR Pell1QR
7167, 70mpbird 240 . . . . . . . . . 10 NN Pell1QR Pell1QR
72 simprr 774 . . . . . . . . . . 11 NN Pell14QR
73 simpll 768 . . . . . . . . . . . . 13 NN Pell14QR NN
74 simplr 770 . . . . . . . . . . . . 13 NN Pell14QR Pell14QR
75 simprl 772 . . . . . . . . . . . . 13 NN Pell14QR
76 pell14qrgapw 35793 . . . . . . . . . . . . 13 NN Pell14QR
7773, 74, 75, 76syl3anc 1292 . . . . . . . . . . . 12 NN Pell14QR
78 pell14qrre 35774 . . . . . . . . . . . . . 14 NN Pell14QR
7978adantr 472 . . . . . . . . . . . . 13 NN Pell14QR
80 ltnsym 9750 . . . . . . . . . . . . 13
811, 79, 80sylancr 676 . . . . . . . . . . . 12 NN Pell14QR
8277, 81mpd 15 . . . . . . . . . . 11 NN Pell14QR
8372, 82pm2.21dd 179 . . . . . . . . . 10 NN Pell14QR PellFund Pell1QR
8447, 54, 66, 71, 83syl22anc 1293 . . . . . . . . 9 NN Pell1QR Pell1QR PellFund Pell1QR
8524, 25, 26, 27, 46, 84syl122anc 1301 . . . . . . . 8 NN Pell1QR PellFund PellFund Pell1QR PellFund PellFund Pell1QR
86 simpll 768 . . . . . . . . 9 NN Pell1QR PellFund PellFund NN
8722adantr 472 . . . . . . . . 9 NN Pell1QR PellFund PellFund
88 simprl 772 . . . . . . . . 9 NN Pell1QR PellFund PellFund PellFund
89 pellfundglb 35804 . . . . . . . . 9 NN PellFund Pell1QRPellFund
9086, 87, 88, 89syl3anc 1292 . . . . . . . 8 NN Pell1QR PellFund PellFund Pell1QRPellFund
9185, 90r19.29a 2918 . . . . . . 7 NN Pell1QR PellFund PellFund PellFund Pell1QR
9291exp32 616 . . . . . 6 NN Pell1QR PellFund PellFund PellFund Pell1QR
93 simp2 1031 . . . . . . . 8 NN Pell1QR PellFund PellFund PellFund
94 simp1r 1055 . . . . . . . 8 NN Pell1QR PellFund PellFund Pell1QR
9593, 94eqeltrd 2549 . . . . . . 7 NN Pell1QR PellFund PellFund PellFund Pell1QR
96953exp 1230 . . . . . 6 NN Pell1QR PellFund PellFund PellFund Pell1QR
9792, 96jaod 387 . . . . 5 NN Pell1QR PellFund PellFund PellFund PellFund Pell1QR
9823, 97sylbid 223 . . . 4 NN Pell1QR PellFund PellFund PellFund Pell1QR
9998impd 438 . . 3 NN Pell1QR PellFund PellFund PellFund Pell1QR
10099rexlimdva 2871 . 2 NN Pell1QRPellFund PellFund PellFund Pell1QR
10117, 100mpd 15 1 NN PellFund Pell1QR
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wo 375   wa 376   w3a 1007   wceq 1452   wcel 1904  wrex 2757   cdif 3387   wss 3390   class class class wbr 4395  cfv 5589  (class class class)co 6308  cr 9556  cc0 9557  c1 9558   caddc 9560   cmul 9562   clt 9693   cle 9694   cdiv 10291  cn 10631  c2 10681  ◻NNcsquarenn 35751  Pell1QRcpell1qr 35752  Pell14QRcpell14qr 35754  PellFundcpellfund 35755 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-ico 11666  df-fz 11811  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-numer 14763  df-denom 14764  df-squarenn 35757  df-pell1qr 35758  df-pell14qr 35759  df-pell1234qr 35760  df-pellfund 35761 This theorem is referenced by:  pellfund14  35817  pellfund14b  35818
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