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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > irrapx1 | Structured version Visualization version GIF version |
Description: Dirichlet's approximation theorem. Every positive irrational number has infinitely many rational approximations which are closer than the inverse squares of their reduced denominators. Lemma 61 in [vandenDries] p. 42. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
Ref | Expression |
---|---|
irrapx1 | β’ (π΄ β (β+ β β) β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qnnen 16160 | . . . 4 β’ β β β | |
2 | nnenom 13949 | . . . 4 β’ β β Ο | |
3 | 1, 2 | entri 9006 | . . 3 β’ β β Ο |
4 | 3, 2 | pm3.2i 469 | . 2 β’ (β β Ο β§ β β Ο) |
5 | ssrab2 4076 | . . . . . 6 β’ {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β β | |
6 | qssre 12947 | . . . . . 6 β’ β β β | |
7 | 5, 6 | sstri 3990 | . . . . 5 β’ {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β β |
8 | 7 | a1i 11 | . . . 4 β’ (π΄ β (β+ β β) β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β β) |
9 | eldifi 4125 | . . . . 5 β’ (π΄ β (β+ β β) β π΄ β β+) | |
10 | 9 | rpred 13020 | . . . 4 β’ (π΄ β (β+ β β) β π΄ β β) |
11 | eldifn 4126 | . . . . 5 β’ (π΄ β (β+ β β) β Β¬ π΄ β β) | |
12 | elrabi 3676 | . . . . 5 β’ (π΄ β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β π΄ β β) | |
13 | 11, 12 | nsyl 140 | . . . 4 β’ (π΄ β (β+ β β) β Β¬ π΄ β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))}) |
14 | irrapxlem6 41867 | . . . . . 6 β’ ((π΄ β β+ β§ π β β+) β βπ β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} (absβ(π β π΄)) < π) | |
15 | 9, 14 | sylan 578 | . . . . 5 β’ ((π΄ β (β+ β β) β§ π β β+) β βπ β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} (absβ(π β π΄)) < π) |
16 | 15 | ralrimiva 3144 | . . . 4 β’ (π΄ β (β+ β β) β βπ β β+ βπ β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} (absβ(π β π΄)) < π) |
17 | rencldnfi 41861 | . . . 4 β’ ((({π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β β β§ π΄ β β β§ Β¬ π΄ β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))}) β§ βπ β β+ βπ β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} (absβ(π β π΄)) < π) β Β¬ {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β Fin) | |
18 | 8, 10, 13, 16, 17 | syl31anc 1371 | . . 3 β’ (π΄ β (β+ β β) β Β¬ {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β Fin) |
19 | 18, 5 | jctil 518 | . 2 β’ (π΄ β (β+ β β) β ({π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β β β§ Β¬ {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β Fin)) |
20 | ctbnfien 41858 | . 2 β’ (((β β Ο β§ β β Ο) β§ ({π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β β β§ Β¬ {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β Fin)) β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β β) | |
21 | 4, 19, 20 | sylancr 585 | 1 β’ (π΄ β (β+ β β) β {π¦ β β β£ (0 < π¦ β§ (absβ(π¦ β π΄)) < ((denomβπ¦)β-2))} β β) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β wcel 2104 βwral 3059 βwrex 3068 {crab 3430 β cdif 3944 β wss 3947 class class class wbr 5147 βcfv 6542 (class class class)co 7411 Οcom 7857 β cen 8938 Fincfn 8941 βcr 11111 0cc0 11112 < clt 11252 β cmin 11448 -cneg 11449 βcn 12216 2c2 12271 βcq 12936 β+crp 12978 βcexp 14031 abscabs 15185 denomcdenom 16674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-q 12937 df-rp 12979 df-ico 13334 df-fz 13489 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-dvds 16202 df-gcd 16440 df-numer 16675 df-denom 16676 |
This theorem is referenced by: pellexlem4 41872 |
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