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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pellfundgt1 | Structured version Visualization version GIF version |
Description: Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
Ref | Expression |
---|---|
pellfundgt1 | β’ (π· β (β β β»NN) β 1 < (PellFundβπ·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 11211 | . 2 β’ (π· β (β β β»NN) β 1 β β) | |
2 | eldifi 4125 | . . . . . . 7 β’ (π· β (β β β»NN) β π· β β) | |
3 | 2 | peano2nnd 12225 | . . . . . 6 β’ (π· β (β β β»NN) β (π· + 1) β β) |
4 | 3 | nnrpd 13010 | . . . . 5 β’ (π· β (β β β»NN) β (π· + 1) β β+) |
5 | 4 | rpsqrtcld 15354 | . . . 4 β’ (π· β (β β β»NN) β (ββ(π· + 1)) β β+) |
6 | 5 | rpred 13012 | . . 3 β’ (π· β (β β β»NN) β (ββ(π· + 1)) β β) |
7 | 2 | nnrpd 13010 | . . . . 5 β’ (π· β (β β β»NN) β π· β β+) |
8 | 7 | rpsqrtcld 15354 | . . . 4 β’ (π· β (β β β»NN) β (ββπ·) β β+) |
9 | 8 | rpred 13012 | . . 3 β’ (π· β (β β β»NN) β (ββπ·) β β) |
10 | 6, 9 | readdcld 11239 | . 2 β’ (π· β (β β β»NN) β ((ββ(π· + 1)) + (ββπ·)) β β) |
11 | pellfundre 41604 | . 2 β’ (π· β (β β β»NN) β (PellFundβπ·) β β) | |
12 | sqrt1 15214 | . . . . 5 β’ (ββ1) = 1 | |
13 | 12, 1 | eqeltrid 2837 | . . . 4 β’ (π· β (β β β»NN) β (ββ1) β β) |
14 | 13, 13 | readdcld 11239 | . . 3 β’ (π· β (β β β»NN) β ((ββ1) + (ββ1)) β β) |
15 | 1lt2 12379 | . . . . 5 β’ 1 < 2 | |
16 | 12, 12 | oveq12i 7417 | . . . . . 6 β’ ((ββ1) + (ββ1)) = (1 + 1) |
17 | 1p1e2 12333 | . . . . . 6 β’ (1 + 1) = 2 | |
18 | 16, 17 | eqtri 2760 | . . . . 5 β’ ((ββ1) + (ββ1)) = 2 |
19 | 15, 18 | breqtrri 5174 | . . . 4 β’ 1 < ((ββ1) + (ββ1)) |
20 | 19 | a1i 11 | . . 3 β’ (π· β (β β β»NN) β 1 < ((ββ1) + (ββ1))) |
21 | 3 | nnge1d 12256 | . . . . 5 β’ (π· β (β β β»NN) β 1 β€ (π· + 1)) |
22 | 0le1 11733 | . . . . . . 7 β’ 0 β€ 1 | |
23 | 22 | a1i 11 | . . . . . 6 β’ (π· β (β β β»NN) β 0 β€ 1) |
24 | 2 | nnred 12223 | . . . . . . 7 β’ (π· β (β β β»NN) β π· β β) |
25 | peano2re 11383 | . . . . . . 7 β’ (π· β β β (π· + 1) β β) | |
26 | 24, 25 | syl 17 | . . . . . 6 β’ (π· β (β β β»NN) β (π· + 1) β β) |
27 | 3 | nnnn0d 12528 | . . . . . . 7 β’ (π· β (β β β»NN) β (π· + 1) β β0) |
28 | 27 | nn0ge0d 12531 | . . . . . 6 β’ (π· β (β β β»NN) β 0 β€ (π· + 1)) |
29 | 1, 23, 26, 28 | sqrtled 15369 | . . . . 5 β’ (π· β (β β β»NN) β (1 β€ (π· + 1) β (ββ1) β€ (ββ(π· + 1)))) |
30 | 21, 29 | mpbid 231 | . . . 4 β’ (π· β (β β β»NN) β (ββ1) β€ (ββ(π· + 1))) |
31 | 2 | nnge1d 12256 | . . . . 5 β’ (π· β (β β β»NN) β 1 β€ π·) |
32 | 2 | nnnn0d 12528 | . . . . . . 7 β’ (π· β (β β β»NN) β π· β β0) |
33 | 32 | nn0ge0d 12531 | . . . . . 6 β’ (π· β (β β β»NN) β 0 β€ π·) |
34 | 1, 23, 24, 33 | sqrtled 15369 | . . . . 5 β’ (π· β (β β β»NN) β (1 β€ π· β (ββ1) β€ (ββπ·))) |
35 | 31, 34 | mpbid 231 | . . . 4 β’ (π· β (β β β»NN) β (ββ1) β€ (ββπ·)) |
36 | 13, 13, 6, 9, 30, 35 | le2addd 11829 | . . 3 β’ (π· β (β β β»NN) β ((ββ1) + (ββ1)) β€ ((ββ(π· + 1)) + (ββπ·))) |
37 | 1, 14, 10, 20, 36 | ltletrd 11370 | . 2 β’ (π· β (β β β»NN) β 1 < ((ββ(π· + 1)) + (ββπ·))) |
38 | pellfundge 41605 | . 2 β’ (π· β (β β β»NN) β ((ββ(π· + 1)) + (ββπ·)) β€ (PellFundβπ·)) | |
39 | 1, 10, 11, 37, 38 | ltletrd 11370 | 1 β’ (π· β (β β β»NN) β 1 < (PellFundβπ·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2106 β cdif 3944 class class class wbr 5147 βcfv 6540 (class class class)co 7405 βcr 11105 0cc0 11106 1c1 11107 + caddc 11109 < clt 11244 β€ cle 11245 βcn 12208 2c2 12263 βcsqrt 15176 β»NNcsquarenn 41559 PellFundcpellfund 41563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-ico 13326 df-fz 13481 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432 df-numer 16667 df-denom 16668 df-squarenn 41564 df-pell1qr 41565 df-pell14qr 41566 df-pell1234qr 41567 df-pellfund 41568 |
This theorem is referenced by: pellfundex 41609 pellfundrp 41611 pellfundne1 41612 pellfund14 41621 |
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