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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 25697 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑎 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 7 | rphalfcl 12942 | . . . . 5 ⊢ (𝑎 ∈ ℝ+ → (𝑎 / 2) ∈ ℝ+) | |
| 8 | 7 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ+) → (𝑎 / 2) ∈ ℝ+) |
| 9 | 7 | ad2antlr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) ∈ ℝ+) |
| 10 | nnrp 12926 | . . . . . . . . . . . 12 ⊢ (𝑞 ∈ ℕ → 𝑞 ∈ ℝ+) | |
| 11 | 10 | ad2antll 727 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑞 ∈ ℝ+) |
| 12 | 3 | nnzd 12526 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 13 | 12 | ad2antrr 724 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑁 ∈ ℤ) |
| 14 | 11, 13 | rpexpcld 14150 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑𝑁) ∈ ℝ+) |
| 15 | 9, 14 | rpdivcld 12974 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) ∈ ℝ+) |
| 16 | 15 | rpred 12957 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑎 / 2) / (𝑞↑𝑁)) ∈ ℝ) |
| 17 | simplr 767 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ+) | |
| 18 | 17, 14 | rpdivcld 12974 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ+) |
| 19 | 18 | rpred 12957 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / (𝑞↑𝑁)) ∈ ℝ) |
| 20 | 4 | adantr 481 | . . . . . . . . . . 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 11465 | . . . . . . . . . . 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 11183 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ) |
| 27 | 26 | abscld 15321 | . . . . . . . 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 12957 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) ∈ ℝ) |
| 30 | rpre 12923 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ ℝ+ → 𝑎 ∈ ℝ) | |
| 31 | 30 | ad2antlr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝑎 ∈ ℝ) |
| 32 | rphalflt 12944 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ ℝ+ → (𝑎 / 2) < 𝑎) | |
| 33 | 32 | ad2antlr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ+) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑎 / 2) < 𝑎) |
| 34 | 29, 31, 14, 33 | ltdiv1dd 13014 | . . . . . . . . 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 413 | . . . . . . 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 11247 | . . . . . . 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 965 | . . . . 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 3201 | . . . 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 7364 | . . . . . . . 8 ⊢ (𝑥 = (𝑎 / 2) → (𝑥 / (𝑞↑𝑁)) = ((𝑎 / 2) / (𝑞↑𝑁))) | |
| 42 | 41 | breq1d 5115 | . . . . . . 7 ⊢ (𝑥 = (𝑎 / 2) → ((𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))) ↔ ((𝑎 / 2) / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) |
| 43 | 42 | orbi2d 914 | . . . . . 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 3212 | . . . . 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 3581 | . . . 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 682 | . . 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 3152 | . 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 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 ℤcz 12499 ℚcq 12873 ℝ+crp 12915 ↑cexp 13967 abscabs 15119 Polycply 25545 degcdgr 25548 |
| 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-mulg 18873 df-subg 18925 df-cntz 19097 df-cmn 19564 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-subrg 20220 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-0p 25034 df-limc 25230 df-dv 25231 df-dvn 25232 df-cpn 25233 df-ply 25549 df-idp 25550 df-coe 25551 df-dgr 25552 df-quot 25651 |
| This theorem is referenced by: aaliou2 25700 |
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