![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
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 11163 | . 2 β’ (π· β (β β β»NN) β 1 β β) | |
2 | eldifi 4091 | . . . . . . 7 β’ (π· β (β β β»NN) β π· β β) | |
3 | 2 | peano2nnd 12177 | . . . . . 6 β’ (π· β (β β β»NN) β (π· + 1) β β) |
4 | 3 | nnrpd 12962 | . . . . 5 β’ (π· β (β β β»NN) β (π· + 1) β β+) |
5 | 4 | rpsqrtcld 15303 | . . . 4 β’ (π· β (β β β»NN) β (ββ(π· + 1)) β β+) |
6 | 5 | rpred 12964 | . . 3 β’ (π· β (β β β»NN) β (ββ(π· + 1)) β β) |
7 | 2 | nnrpd 12962 | . . . . 5 β’ (π· β (β β β»NN) β π· β β+) |
8 | 7 | rpsqrtcld 15303 | . . . 4 β’ (π· β (β β β»NN) β (ββπ·) β β+) |
9 | 8 | rpred 12964 | . . 3 β’ (π· β (β β β»NN) β (ββπ·) β β) |
10 | 6, 9 | readdcld 11191 | . 2 β’ (π· β (β β β»NN) β ((ββ(π· + 1)) + (ββπ·)) β β) |
11 | pellfundre 41233 | . 2 β’ (π· β (β β β»NN) β (PellFundβπ·) β β) | |
12 | sqrt1 15163 | . . . . 5 β’ (ββ1) = 1 | |
13 | 12, 1 | eqeltrid 2842 | . . . 4 β’ (π· β (β β β»NN) β (ββ1) β β) |
14 | 13, 13 | readdcld 11191 | . . 3 β’ (π· β (β β β»NN) β ((ββ1) + (ββ1)) β β) |
15 | 1lt2 12331 | . . . . 5 β’ 1 < 2 | |
16 | 12, 12 | oveq12i 7374 | . . . . . 6 β’ ((ββ1) + (ββ1)) = (1 + 1) |
17 | 1p1e2 12285 | . . . . . 6 β’ (1 + 1) = 2 | |
18 | 16, 17 | eqtri 2765 | . . . . 5 β’ ((ββ1) + (ββ1)) = 2 |
19 | 15, 18 | breqtrri 5137 | . . . 4 β’ 1 < ((ββ1) + (ββ1)) |
20 | 19 | a1i 11 | . . 3 β’ (π· β (β β β»NN) β 1 < ((ββ1) + (ββ1))) |
21 | 3 | nnge1d 12208 | . . . . 5 β’ (π· β (β β β»NN) β 1 β€ (π· + 1)) |
22 | 0le1 11685 | . . . . . . 7 β’ 0 β€ 1 | |
23 | 22 | a1i 11 | . . . . . 6 β’ (π· β (β β β»NN) β 0 β€ 1) |
24 | 2 | nnred 12175 | . . . . . . 7 β’ (π· β (β β β»NN) β π· β β) |
25 | peano2re 11335 | . . . . . . 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 12480 | . . . . . . 7 β’ (π· β (β β β»NN) β (π· + 1) β β0) |
28 | 27 | nn0ge0d 12483 | . . . . . 6 β’ (π· β (β β β»NN) β 0 β€ (π· + 1)) |
29 | 1, 23, 26, 28 | sqrtled 15318 | . . . . 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 12208 | . . . . 5 β’ (π· β (β β β»NN) β 1 β€ π·) |
32 | 2 | nnnn0d 12480 | . . . . . . 7 β’ (π· β (β β β»NN) β π· β β0) |
33 | 32 | nn0ge0d 12483 | . . . . . 6 β’ (π· β (β β β»NN) β 0 β€ π·) |
34 | 1, 23, 24, 33 | sqrtled 15318 | . . . . 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 11781 | . . 3 β’ (π· β (β β β»NN) β ((ββ1) + (ββ1)) β€ ((ββ(π· + 1)) + (ββπ·))) |
37 | 1, 14, 10, 20, 36 | ltletrd 11322 | . 2 β’ (π· β (β β β»NN) β 1 < ((ββ(π· + 1)) + (ββπ·))) |
38 | pellfundge 41234 | . 2 β’ (π· β (β β β»NN) β ((ββ(π· + 1)) + (ββπ·)) β€ (PellFundβπ·)) | |
39 | 1, 10, 11, 37, 38 | ltletrd 11322 | 1 β’ (π· β (β β β»NN) β 1 < (PellFundβπ·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 β cdif 3912 class class class wbr 5110 βcfv 6501 (class class class)co 7362 βcr 11057 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 β€ cle 11197 βcn 12160 2c2 12215 βcsqrt 15125 β»NNcsquarenn 41188 PellFundcpellfund 41192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-omul 8422 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-acn 9885 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-ico 13277 df-fz 13432 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-dvds 16144 df-gcd 16382 df-numer 16617 df-denom 16618 df-squarenn 41193 df-pell1qr 41194 df-pell14qr 41195 df-pell1234qr 41196 df-pellfund 41197 |
This theorem is referenced by: pellfundex 41238 pellfundrp 41240 pellfundne1 41241 pellfund14 41250 |
Copyright terms: Public domain | W3C validator |