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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 | ⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ≈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qnnen 15148 | . . . 4 ⊢ ℚ ≈ ℕ | |
2 | nnenom 12987 | . . . 4 ⊢ ℕ ≈ ω | |
3 | 1, 2 | entri 8167 | . . 3 ⊢ ℚ ≈ ω |
4 | 3, 2 | pm3.2i 456 | . 2 ⊢ (ℚ ≈ ω ∧ ℕ ≈ ω) |
5 | ssrab2 3836 | . . . . . 6 ⊢ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ⊆ ℚ | |
6 | qssre 12006 | . . . . . 6 ⊢ ℚ ⊆ ℝ | |
7 | 5, 6 | sstri 3761 | . . . . 5 ⊢ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ⊆ ℝ |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ⊆ ℝ) |
9 | eldifi 3883 | . . . . 5 ⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → 𝐴 ∈ ℝ+) | |
10 | 9 | rpred 12075 | . . . 4 ⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → 𝐴 ∈ ℝ) |
11 | eldifn 3884 | . . . . 5 ⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → ¬ 𝐴 ∈ ℚ) | |
12 | elrabi 3510 | . . . . 5 ⊢ (𝐴 ∈ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} → 𝐴 ∈ ℚ) | |
13 | 11, 12 | nsyl 137 | . . . 4 ⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → ¬ 𝐴 ∈ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))}) |
14 | irrapxlem6 37915 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑎 ∈ ℝ+) → ∃𝑏 ∈ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} (abs‘(𝑏 − 𝐴)) < 𝑎) | |
15 | 9, 14 | sylan 569 | . . . . 5 ⊢ ((𝐴 ∈ (ℝ+ ∖ ℚ) ∧ 𝑎 ∈ ℝ+) → ∃𝑏 ∈ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} (abs‘(𝑏 − 𝐴)) < 𝑎) |
16 | 15 | ralrimiva 3115 | . . . 4 ⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → ∀𝑎 ∈ ℝ+ ∃𝑏 ∈ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} (abs‘(𝑏 − 𝐴)) < 𝑎) |
17 | rencldnfi 37909 | . . . 4 ⊢ ((({𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ⊆ ℝ ∧ 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))}) ∧ ∀𝑎 ∈ ℝ+ ∃𝑏 ∈ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} (abs‘(𝑏 − 𝐴)) < 𝑎) → ¬ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ∈ Fin) | |
18 | 8, 10, 13, 16, 17 | syl31anc 1479 | . . 3 ⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → ¬ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ∈ Fin) |
19 | 18, 5 | jctil 509 | . 2 ⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → ({𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ⊆ ℚ ∧ ¬ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ∈ Fin)) |
20 | ctbnfien 37906 | . 2 ⊢ (((ℚ ≈ ω ∧ ℕ ≈ ω) ∧ ({𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ⊆ ℚ ∧ ¬ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ∈ Fin)) → {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ≈ ℕ) | |
21 | 4, 19, 20 | sylancr 575 | 1 ⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ≈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 ∈ wcel 2145 ∀wral 3061 ∃wrex 3062 {crab 3065 ∖ cdif 3720 ⊆ wss 3723 class class class wbr 4787 ‘cfv 6030 (class class class)co 6796 ωcom 7216 ≈ cen 8110 Fincfn 8113 ℝcr 10141 0cc0 10142 < clt 10280 − cmin 10472 -cneg 10473 ℕcn 11226 2c2 11276 ℚcq 11996 ℝ+crp 12035 ↑cexp 13067 abscabs 14182 denomcdenom 15649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-omul 7722 df-er 7900 df-map 8015 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-sup 8508 df-inf 8509 df-oi 8575 df-card 8969 df-acn 8972 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-n0 11500 df-xnn0 11571 df-z 11585 df-uz 11894 df-q 11997 df-rp 12036 df-ico 12386 df-fz 12534 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-dvds 15190 df-gcd 15425 df-numer 15650 df-denom 15651 |
This theorem is referenced by: pellexlem4 37920 |
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