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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 26313 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 7 | rphalfcl 12946 | . . . . 5 ⊢ (𝑎 ∈ ℝ+ → (𝑎 / 2) ∈ ℝ+) | |
| 8 | 7 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ+) → (𝑎 / 2) ∈ ℝ+) |
| 9 | 7 | ad2antlr 728 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) ∈ ℝ+) |
| 10 | nnrp 12929 | . . . . . . . . . . . 12 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℝ+) | |
| 11 | 10 | ad2antll 730 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑞 ∈ ℝ+) |
| 12 | 3 | nnzd 12526 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 13 | 12 | ad2antrr 727 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℤ) |
| 14 | 11, 13 | rpexpcld 14182 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℝ+) |
| 15 | 9, 14 | rpdivcld 12978 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) ∈ ℝ+) |
| 16 | 15 | rpred 12961 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) ∈ ℝ) |
| 17 | simplr 769 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ+) | |
| 18 | 17, 14 | rpdivcld 12978 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ+) |
| 19 | 18 | rpred 12961 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ) |
| 20 | 4 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ+) → 𝐴 ∈ ℝ) |
| 21 | znq 12877 | . . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℚ) | |
| 22 | qre 12878 | . . . . . . . . . . . 12 ⊢ ((𝑝 / 𝑞) ∈ ℚ → (𝑝 / 𝑞) ∈ ℝ) | |
| 23 | 21, 22 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℝ) |
| 24 | resubcl 11457 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝑝 / 𝑞) ∈ ℝ) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ) | |
| 25 | 20, 23, 24 | syl2an 597 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ) |
| 26 | 25 | recnd 11172 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ) |
| 27 | 26 | abscld 15374 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ) |
| 28 | 16, 19, 27 | 3jca 1129 | . . . . . . 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 12961 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) ∈ ℝ) |
| 30 | rpre 12926 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ ℝ+ → 𝑎 ∈ ℝ) | |
| 31 | 30 | ad2antlr 728 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ) |
| 32 | rphalflt 12948 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ ℝ+ → (𝑎 / 2) < 𝑎) | |
| 33 | 32 | ad2antlr 728 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) < 𝑎) |
| 34 | 29, 31, 14, 33 | ltdiv1dd 13018 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) < (𝑎 / (𝑞↑𝑁))) |
| 35 | 34 | anim1i 616 | . . . . . . . 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 412 | . . . . . . 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 11237 | . . . . . . 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 969 | . . . . 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 3185 | . . . 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 7375 | . . . . . . . 8 ⊢ (𝑥 = (𝑎 / 2) → (𝑥 / (𝑞↑𝑁)) = ((𝑎 / 2) / (𝑞↑𝑁))) | |
| 42 | 41 | breq1d 5110 | . . . . . . 7 ⊢ (𝑥 = (𝑎 / 2) → ((𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))) ↔ ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 43 | 42 | orbi2d 916 | . . . . . 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 3202 | . . . . 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 3578 | . . . 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 685 | . . 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 3139 | . 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 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 ℝcr 11037 0cc0 11038 < clt 11178 ≤ cle 11179 − cmin 11376 / cdiv 11806 ℕcn 12157 2c2 12212 ℤcz 12500 ℚcq 12873 ℝ+crp 12917 ↑cexp 13996 abscabs 15169 Polycply 26157 degcdgr 26160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-rlim 15424 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-mulg 19010 df-subg 19065 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-subrng 20491 df-subrg 20515 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-cmp 23343 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-0p 25639 df-limc 25835 df-dv 25836 df-dvn 25837 df-cpn 25838 df-ply 26161 df-idp 26162 df-coe 26163 df-dgr 26164 df-quot 26267 |
| This theorem is referenced by: aaliou2 26316 |
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