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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 11231 | . 2 β’ (π· β (β β β»NN) β 1 β β) | |
2 | eldifi 4122 | . . . . . . 7 β’ (π· β (β β β»NN) β π· β β) | |
3 | 2 | peano2nnd 12245 | . . . . . 6 β’ (π· β (β β β»NN) β (π· + 1) β β) |
4 | 3 | nnrpd 13032 | . . . . 5 β’ (π· β (β β β»NN) β (π· + 1) β β+) |
5 | 4 | rpsqrtcld 15376 | . . . 4 β’ (π· β (β β β»NN) β (ββ(π· + 1)) β β+) |
6 | 5 | rpred 13034 | . . 3 β’ (π· β (β β β»NN) β (ββ(π· + 1)) β β) |
7 | 2 | nnrpd 13032 | . . . . 5 β’ (π· β (β β β»NN) β π· β β+) |
8 | 7 | rpsqrtcld 15376 | . . . 4 β’ (π· β (β β β»NN) β (ββπ·) β β+) |
9 | 8 | rpred 13034 | . . 3 β’ (π· β (β β β»NN) β (ββπ·) β β) |
10 | 6, 9 | readdcld 11259 | . 2 β’ (π· β (β β β»NN) β ((ββ(π· + 1)) + (ββπ·)) β β) |
11 | pellfundre 42213 | . 2 β’ (π· β (β β β»NN) β (PellFundβπ·) β β) | |
12 | sqrt1 15236 | . . . . 5 β’ (ββ1) = 1 | |
13 | 12, 1 | eqeltrid 2832 | . . . 4 β’ (π· β (β β β»NN) β (ββ1) β β) |
14 | 13, 13 | readdcld 11259 | . . 3 β’ (π· β (β β β»NN) β ((ββ1) + (ββ1)) β β) |
15 | 1lt2 12399 | . . . . 5 β’ 1 < 2 | |
16 | 12, 12 | oveq12i 7426 | . . . . . 6 β’ ((ββ1) + (ββ1)) = (1 + 1) |
17 | 1p1e2 12353 | . . . . . 6 β’ (1 + 1) = 2 | |
18 | 16, 17 | eqtri 2755 | . . . . 5 β’ ((ββ1) + (ββ1)) = 2 |
19 | 15, 18 | breqtrri 5169 | . . . 4 β’ 1 < ((ββ1) + (ββ1)) |
20 | 19 | a1i 11 | . . 3 β’ (π· β (β β β»NN) β 1 < ((ββ1) + (ββ1))) |
21 | 3 | nnge1d 12276 | . . . . 5 β’ (π· β (β β β»NN) β 1 β€ (π· + 1)) |
22 | 0le1 11753 | . . . . . . 7 β’ 0 β€ 1 | |
23 | 22 | a1i 11 | . . . . . 6 β’ (π· β (β β β»NN) β 0 β€ 1) |
24 | 2 | nnred 12243 | . . . . . . 7 β’ (π· β (β β β»NN) β π· β β) |
25 | peano2re 11403 | . . . . . . 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 12548 | . . . . . . 7 β’ (π· β (β β β»NN) β (π· + 1) β β0) |
28 | 27 | nn0ge0d 12551 | . . . . . 6 β’ (π· β (β β β»NN) β 0 β€ (π· + 1)) |
29 | 1, 23, 26, 28 | sqrtled 15391 | . . . . 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 12276 | . . . . 5 β’ (π· β (β β β»NN) β 1 β€ π·) |
32 | 2 | nnnn0d 12548 | . . . . . . 7 β’ (π· β (β β β»NN) β π· β β0) |
33 | 32 | nn0ge0d 12551 | . . . . . 6 β’ (π· β (β β β»NN) β 0 β€ π·) |
34 | 1, 23, 24, 33 | sqrtled 15391 | . . . . 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 11849 | . . 3 β’ (π· β (β β β»NN) β ((ββ1) + (ββ1)) β€ ((ββ(π· + 1)) + (ββπ·))) |
37 | 1, 14, 10, 20, 36 | ltletrd 11390 | . 2 β’ (π· β (β β β»NN) β 1 < ((ββ(π· + 1)) + (ββπ·))) |
38 | pellfundge 42214 | . 2 β’ (π· β (β β β»NN) β ((ββ(π· + 1)) + (ββπ·)) β€ (PellFundβπ·)) | |
39 | 1, 10, 11, 37, 38 | ltletrd 11390 | 1 β’ (π· β (β β β»NN) β 1 < (PellFundβπ·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2099 β cdif 3941 class class class wbr 5142 βcfv 6542 (class class class)co 7414 βcr 11123 0cc0 11124 1c1 11125 + caddc 11127 < clt 11264 β€ cle 11265 βcn 12228 2c2 12283 βcsqrt 15198 β»NNcsquarenn 42168 PellFundcpellfund 42172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-oadd 8482 df-omul 8483 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-oi 9519 df-card 9948 df-acn 9951 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-xnn0 12561 df-z 12575 df-uz 12839 df-q 12949 df-rp 12993 df-ico 13348 df-fz 13503 df-fl 13775 df-mod 13853 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-dvds 16217 df-gcd 16455 df-numer 16692 df-denom 16693 df-squarenn 42173 df-pell1qr 42174 df-pell14qr 42175 df-pell1234qr 42176 df-pellfund 42177 |
This theorem is referenced by: pellfundex 42218 pellfundrp 42220 pellfundne1 42221 pellfund14 42230 |
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