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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 26252 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 7 | rphalfcl 12987 | . . . . 5 ⊢ (𝑎 ∈ ℝ+ → (𝑎 / 2) ∈ ℝ+) | |
| 8 | 7 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ+) → (𝑎 / 2) ∈ ℝ+) |
| 9 | 7 | ad2antlr 727 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) ∈ ℝ+) |
| 10 | nnrp 12970 | . . . . . . . . . . . 12 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℝ+) | |
| 11 | 10 | ad2antll 729 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑞 ∈ ℝ+) |
| 12 | 3 | nnzd 12563 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 13 | 12 | ad2antrr 726 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℤ) |
| 14 | 11, 13 | rpexpcld 14219 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℝ+) |
| 15 | 9, 14 | rpdivcld 13019 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) ∈ ℝ+) |
| 16 | 15 | rpred 13002 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) ∈ ℝ) |
| 17 | simplr 768 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ+) | |
| 18 | 17, 14 | rpdivcld 13019 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ+) |
| 19 | 18 | rpred 13002 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ) |
| 20 | 4 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ+) → 𝐴 ∈ ℝ) |
| 21 | znq 12918 | . . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℚ) | |
| 22 | qre 12919 | . . . . . . . . . . . 12 ⊢ ((𝑝 / 𝑞) ∈ ℚ → (𝑝 / 𝑞) ∈ ℝ) | |
| 23 | 21, 22 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℝ) |
| 24 | resubcl 11493 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝑝 / 𝑞) ∈ ℝ) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ) | |
| 25 | 20, 23, 24 | syl2an 596 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℝ) |
| 26 | 25 | recnd 11209 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ) |
| 27 | 26 | abscld 15412 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ) |
| 28 | 16, 19, 27 | 3jca 1128 | . . . . . . 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 13002 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) ∈ ℝ) |
| 30 | rpre 12967 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ ℝ+ → 𝑎 ∈ ℝ) | |
| 31 | 30 | ad2antlr 727 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ) |
| 32 | rphalflt 12989 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ ℝ+ → (𝑎 / 2) < 𝑎) | |
| 33 | 32 | ad2antlr 727 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) < 𝑎) |
| 34 | 29, 31, 14, 33 | ltdiv1dd 13059 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) < (𝑎 / (𝑞↑𝑁))) |
| 35 | 34 | anim1i 615 | . . . . . . . 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 11273 | . . . . . . 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 968 | . . . . 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 7397 | . . . . . . . 8 ⊢ (𝑥 = (𝑎 / 2) → (𝑥 / (𝑞↑𝑁)) = ((𝑎 / 2) / (𝑞↑𝑁))) | |
| 42 | 41 | breq1d 5120 | . . . . . . 7 ⊢ (𝑥 = (𝑎 / 2) → ((𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))) ↔ ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 43 | 42 | orbi2d 915 | . . . . . 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 3591 | . . . 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 684 | . . 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 3135 | . 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 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 class class class wbr 5110 ‘cfv 6514 (class class class)co 7390 ℝcr 11074 0cc0 11075 < clt 11215 ≤ cle 11216 − cmin 11412 / cdiv 11842 ℕcn 12193 2c2 12248 ℤcz 12536 ℚcq 12914 ℝ+crp 12958 ↑cexp 14033 abscabs 15207 Polycply 26096 degcdgr 26099 |
| 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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-xnn0 12523 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-mulg 19007 df-subg 19062 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-subrng 20462 df-subrg 20486 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-cmp 23281 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-0p 25578 df-limc 25774 df-dv 25775 df-dvn 25776 df-cpn 25777 df-ply 26100 df-idp 26101 df-coe 26102 df-dgr 26103 df-quot 26206 |
| This theorem is referenced by: aaliou2 26255 |
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