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| Mirrors > Home > MPE Home > Th. List > aaliou | Structured version Visualization version GIF version | ||
| Description: Liouville's theorem on diophantine approximation: Any algebraic number, being a root of a polynomial 𝐹 in integer coefficients, is not approximable beyond order 𝑁 = deg(𝐹) by rational numbers. In this form, it also applies to rational numbers themselves, which are not well approximable by other rational numbers. This is Metamath 100 proof #18. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
| Ref | Expression |
|---|---|
| aalioulem2.a | ⊢ 𝑁 = (deg‘𝐹) |
| aalioulem2.b | ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ)) |
| aalioulem2.c | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| aalioulem2.d | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| aalioulem3.e | ⊢ (𝜑 → (𝐹‘𝐴) = 0) |
| Ref | Expression |
|---|---|
| aaliou | ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | aalioulem2.a | . . 3 ⊢ 𝑁 = (deg‘𝐹) | |
| 2 | aalioulem2.b | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ)) | |
| 3 | aalioulem2.c | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 4 | aalioulem2.d | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 5 | aalioulem3.e | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = 0) | |
| 6 | 1, 2, 3, 4, 5 | aalioulem6 24311 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 7 | rphalfcl 12060 | . . . . 5 ⊢ (𝑎 ∈ ℝ+ → (𝑎 / 2) ∈ ℝ+) | |
| 8 | 7 | adantl 467 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ+) → (𝑎 / 2) ∈ ℝ+) |
| 9 | 7 | ad2antlr 706 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) ∈ ℝ+) |
| 10 | nnrp 12044 | . . . . . . . . . . . 12 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℝ+) | |
| 11 | 10 | ad2antll 708 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑞 ∈ ℝ+) |
| 12 | 3 | nnzd 11687 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 13 | 12 | ad2antrr 705 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℤ) |
| 14 | 11, 13 | rpexpcld 13238 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℝ+) |
| 15 | 9, 14 | rpdivcld 12091 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) ∈ ℝ+) |
| 16 | 15 | rpred 12074 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) ∈ ℝ) |
| 17 | simplr 752 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ+) | |
| 18 | 17, 14 | rpdivcld 12091 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ+) |
| 19 | 18 | rpred 12074 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ) |
| 20 | 4 | adantr 466 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ+) → 𝐴 ∈ ℝ) |
| 21 | znq 11999 | . . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℚ) | |
| 22 | qre 12000 | . . . . . . . . . . . 12 ⊢ ((𝑝 / 𝑞) ∈ ℚ → (𝑝 / 𝑞) ∈ ℝ) | |
| 23 | 21, 22 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℝ) |
| 24 | resubcl 10550 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝑝 / 𝑞) ∈ ℝ) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ) | |
| 25 | 20, 23, 24 | syl2an 583 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ) |
| 26 | 25 | recnd 10273 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ) |
| 27 | 26 | abscld 14382 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ) |
| 28 | 16, 19, 27 | 3jca 1122 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((𝑎 / 2) / (𝑞↑𝑁)) ∈ ℝ ∧ (𝑎 / (𝑞↑𝑁)) ∈ ℝ ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ)) |
| 29 | 9 | rpred 12074 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) ∈ ℝ) |
| 30 | rpre 12041 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ ℝ+ → 𝑎 ∈ ℝ) | |
| 31 | 30 | ad2antlr 706 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ) |
| 32 | rphalflt 12062 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ ℝ+ → (𝑎 / 2) < 𝑎) | |
| 33 | 32 | ad2antlr 706 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) < 𝑎) |
| 34 | 29, 31, 14, 33 | ltdiv1dd 12131 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) < (𝑎 / (𝑞↑𝑁))) |
| 35 | 34 | anim1i 602 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) ∧ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (((𝑎 / 2) / (𝑞↑𝑁)) < (𝑎 / (𝑞↑𝑁)) ∧ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 36 | 35 | ex 397 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))) → (((𝑎 / 2) / (𝑞↑𝑁)) < (𝑎 / (𝑞↑𝑁)) ∧ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) |
| 37 | ltletr 10334 | . . . . . . 7 ⊢ ((((𝑎 / 2) / (𝑞↑𝑁)) ∈ ℝ ∧ (𝑎 / (𝑞↑𝑁)) ∈ ℝ ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ) → ((((𝑎 / 2) / (𝑞↑𝑁)) < (𝑎 / (𝑞↑𝑁)) ∧ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) | |
| 38 | 28, 36, 37 | sylsyld 61 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))) → ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 39 | 38 | orim2d 951 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → (𝐴 = (𝑝 / 𝑞) ∨ ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))) |
| 40 | 39 | ralimdvva 3113 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ+) → (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))) |
| 41 | oveq1 6802 | . . . . . . . 8 ⊢ (𝑥 = (𝑎 / 2) → (𝑥 / (𝑞↑𝑁)) = ((𝑎 / 2) / (𝑞↑𝑁))) | |
| 42 | 41 | breq1d 4797 | . . . . . . 7 ⊢ (𝑥 = (𝑎 / 2) → ((𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))) ↔ ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 43 | 42 | orbi2d 901 | . . . . . 6 ⊢ (𝑥 = (𝑎 / 2) → ((𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ (𝐴 = (𝑝 / 𝑞) ∨ ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))) |
| 44 | 43 | 2ralbidv 3138 | . . . . 5 ⊢ (𝑥 = (𝑎 / 2) → (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))) |
| 45 | 44 | rspcev 3460 | . . . 4 ⊢ (((𝑎 / 2) ∈ ℝ+ ∧ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 46 | 8, 40, 45 | syl6an 663 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ+) → (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))) |
| 47 | 46 | rexlimdva 3179 | . 2 ⊢ (𝜑 → (∃𝑎 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))) → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))) |
| 48 | 6, 47 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 382 ∨ wo 836 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ∃wrex 3062 class class class wbr 4787 ‘cfv 6030 (class class class)co 6795 ℝcr 10140 0cc0 10141 < clt 10279 ≤ cle 10280 − cmin 10471 / cdiv 10889 ℕcn 11225 2c2 11275 ℤcz 11583 ℚcq 11995 ℝ+crp 12034 ↑cexp 13066 abscabs 14181 Polycply 24159 degcdgr 24162 |
| 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 7099 ax-inf2 8705 ax-cnex 10197 ax-resscn 10198 ax-1cn 10199 ax-icn 10200 ax-addcl 10201 ax-addrcl 10202 ax-mulcl 10203 ax-mulrcl 10204 ax-mulcom 10205 ax-addass 10206 ax-mulass 10207 ax-distr 10208 ax-i2m1 10209 ax-1ne0 10210 ax-1rid 10211 ax-rnegex 10212 ax-rrecex 10213 ax-cnre 10214 ax-pre-lttri 10215 ax-pre-lttrn 10216 ax-pre-ltadd 10217 ax-pre-mulgt0 10218 ax-pre-sup 10219 ax-addf 10220 ax-mulf 10221 |
| This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 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-iin 4658 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 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-of 7047 df-om 7216 df-1st 7318 df-2nd 7319 df-supp 7450 df-wrecs 7562 df-recs 7624 df-rdg 7662 df-1o 7716 df-2o 7717 df-oadd 7720 df-er 7899 df-map 8014 df-pm 8015 df-ixp 8066 df-en 8113 df-dom 8114 df-sdom 8115 df-fin 8116 df-fsupp 8435 df-fi 8476 df-sup 8507 df-inf 8508 df-oi 8574 df-card 8968 df-cda 9195 df-pnf 10281 df-mnf 10282 df-xr 10283 df-ltxr 10284 df-le 10285 df-sub 10473 df-neg 10474 df-div 10890 df-nn 11226 df-2 11284 df-3 11285 df-4 11286 df-5 11287 df-6 11288 df-7 11289 df-8 11290 df-9 11291 df-n0 11499 df-xnn0 11570 df-z 11584 df-dec 11700 df-uz 11893 df-q 11996 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-ico 12385 df-icc 12386 df-fz 12533 df-fzo 12673 df-fl 12800 df-seq 13008 df-exp 13067 df-hash 13321 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-clim 14426 df-rlim 14427 df-sum 14624 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-sca 16164 df-vsca 16165 df-ip 16166 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-hom 16173 df-cco 16174 df-rest 16290 df-topn 16291 df-0g 16309 df-gsum 16310 df-topgen 16311 df-pt 16312 df-prds 16315 df-xrs 16369 df-qtop 16374 df-imas 16375 df-xps 16377 df-mre 16453 df-mrc 16454 df-acs 16456 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-grp 17632 df-minusg 17633 df-mulg 17748 df-subg 17798 df-cntz 17956 df-cmn 18401 df-mgp 18697 df-ur 18709 df-ring 18756 df-cring 18757 df-subrg 18987 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-fbas 19957 df-fg 19958 df-cnfld 19961 df-top 20918 df-topon 20935 df-topsp 20957 df-bases 20970 df-cld 21043 df-ntr 21044 df-cls 21045 df-nei 21122 df-lp 21160 df-perf 21161 df-cn 21251 df-cnp 21252 df-haus 21339 df-cmp 21410 df-tx 21585 df-hmeo 21778 df-fil 21869 df-fm 21961 df-flim 21962 df-flf 21963 df-xms 22344 df-ms 22345 df-tms 22346 df-cncf 22900 df-0p 23656 df-limc 23849 df-dv 23850 df-dvn 23851 df-cpn 23852 df-ply 24163 df-idp 24164 df-coe 24165 df-dgr 24166 df-quot 24265 |
| This theorem is referenced by: aaliou2 24314 |
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