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| Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem7 | Structured version Visualization version GIF version | ||
| Description: Lemma for heibor 37828. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014.) |
| Ref | Expression |
|---|---|
| heibor.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
| heibor.3 | ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣} |
| heibor.4 | ⊢ 𝐺 = {〈𝑦, 𝑛〉 ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} |
| heibor.5 | ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) |
| heibor.6 | ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) |
| heibor.7 | ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) |
| heibor.8 | ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) |
| heibor.9 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) |
| heibor.10 | ⊢ (𝜑 → 𝐶𝐺0) |
| heibor.11 | ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) |
| heibor.12 | ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉) |
| Ref | Expression |
|---|---|
| heiborlem7 | ⊢ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3re 12346 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
| 2 | 3pos 12371 | . . . . . . 7 ⊢ 0 < 3 | |
| 3 | 1, 2 | elrpii 13037 | . . . . . 6 ⊢ 3 ∈ ℝ+ |
| 4 | rpdivcl 13060 | . . . . . 6 ⊢ ((𝑟 ∈ ℝ+ ∧ 3 ∈ ℝ+) → (𝑟 / 3) ∈ ℝ+) | |
| 5 | 3, 4 | mpan2 691 | . . . . 5 ⊢ (𝑟 ∈ ℝ+ → (𝑟 / 3) ∈ ℝ+) |
| 6 | 2re 12340 | . . . . . 6 ⊢ 2 ∈ ℝ | |
| 7 | 1lt2 12437 | . . . . . 6 ⊢ 1 < 2 | |
| 8 | expnlbnd 14272 | . . . . . 6 ⊢ (((𝑟 / 3) ∈ ℝ+ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑘 ∈ ℕ (1 / (2↑𝑘)) < (𝑟 / 3)) | |
| 9 | 6, 7, 8 | mp3an23 1455 | . . . . 5 ⊢ ((𝑟 / 3) ∈ ℝ+ → ∃𝑘 ∈ ℕ (1 / (2↑𝑘)) < (𝑟 / 3)) |
| 10 | 5, 9 | syl 17 | . . . 4 ⊢ (𝑟 ∈ ℝ+ → ∃𝑘 ∈ ℕ (1 / (2↑𝑘)) < (𝑟 / 3)) |
| 11 | 2nn 12339 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ | |
| 12 | nnnn0 12533 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
| 13 | nnexpcl 14115 | . . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ) | |
| 14 | 11, 12, 13 | sylancr 587 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℕ) |
| 15 | 14 | nnrpd 13075 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℝ+) |
| 16 | rpcn 13045 | . . . . . . . . . 10 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ∈ ℂ) | |
| 17 | rpne0 13051 | . . . . . . . . . 10 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ≠ 0) | |
| 18 | 3cn 12347 | . . . . . . . . . . 11 ⊢ 3 ∈ ℂ | |
| 19 | divrec 11938 | . . . . . . . . . . 11 ⊢ ((3 ∈ ℂ ∧ (2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) | |
| 20 | 18, 19 | mp3an1 1450 | . . . . . . . . . 10 ⊢ (((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) |
| 21 | 16, 17, 20 | syl2anc 584 | . . . . . . . . 9 ⊢ ((2↑𝑘) ∈ ℝ+ → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) |
| 22 | 15, 21 | syl 17 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) |
| 23 | 22 | adantl 481 | . . . . . . 7 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) |
| 24 | 23 | breq1d 5153 | . . . . . 6 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → ((3 / (2↑𝑘)) < 𝑟 ↔ (3 · (1 / (2↑𝑘))) < 𝑟)) |
| 25 | 14 | nnrecred 12317 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ∈ ℝ) |
| 26 | rpre 13043 | . . . . . . 7 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ) | |
| 27 | 1, 2 | pm3.2i 470 | . . . . . . . 8 ⊢ (3 ∈ ℝ ∧ 0 < 3) |
| 28 | ltmuldiv2 12142 | . . . . . . . 8 ⊢ (((1 / (2↑𝑘)) ∈ ℝ ∧ 𝑟 ∈ ℝ ∧ (3 ∈ ℝ ∧ 0 < 3)) → ((3 · (1 / (2↑𝑘))) < 𝑟 ↔ (1 / (2↑𝑘)) < (𝑟 / 3))) | |
| 29 | 27, 28 | mp3an3 1452 | . . . . . . 7 ⊢ (((1 / (2↑𝑘)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((3 · (1 / (2↑𝑘))) < 𝑟 ↔ (1 / (2↑𝑘)) < (𝑟 / 3))) |
| 30 | 25, 26, 29 | syl2anr 597 | . . . . . 6 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → ((3 · (1 / (2↑𝑘))) < 𝑟 ↔ (1 / (2↑𝑘)) < (𝑟 / 3))) |
| 31 | 24, 30 | bitrd 279 | . . . . 5 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → ((3 / (2↑𝑘)) < 𝑟 ↔ (1 / (2↑𝑘)) < (𝑟 / 3))) |
| 32 | 31 | rexbidva 3177 | . . . 4 ⊢ (𝑟 ∈ ℝ+ → (∃𝑘 ∈ ℕ (3 / (2↑𝑘)) < 𝑟 ↔ ∃𝑘 ∈ ℕ (1 / (2↑𝑘)) < (𝑟 / 3))) |
| 33 | 10, 32 | mpbird 257 | . . 3 ⊢ (𝑟 ∈ ℝ+ → ∃𝑘 ∈ ℕ (3 / (2↑𝑘)) < 𝑟) |
| 34 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑆‘𝑛) = (𝑆‘𝑘)) | |
| 35 | oveq2 7439 | . . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘)) | |
| 36 | 35 | oveq2d 7447 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘))) |
| 37 | 34, 36 | opeq12d 4881 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 = 〈(𝑆‘𝑘), (3 / (2↑𝑘))〉) |
| 38 | heibor.12 | . . . . . . . 8 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉) | |
| 39 | opex 5469 | . . . . . . . 8 ⊢ 〈(𝑆‘𝑘), (3 / (2↑𝑘))〉 ∈ V | |
| 40 | 37, 38, 39 | fvmpt 7016 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑀‘𝑘) = 〈(𝑆‘𝑘), (3 / (2↑𝑘))〉) |
| 41 | 40 | fveq2d 6910 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (2nd ‘(𝑀‘𝑘)) = (2nd ‘〈(𝑆‘𝑘), (3 / (2↑𝑘))〉)) |
| 42 | fvex 6919 | . . . . . . 7 ⊢ (𝑆‘𝑘) ∈ V | |
| 43 | ovex 7464 | . . . . . . 7 ⊢ (3 / (2↑𝑘)) ∈ V | |
| 44 | 42, 43 | op2nd 8023 | . . . . . 6 ⊢ (2nd ‘〈(𝑆‘𝑘), (3 / (2↑𝑘))〉) = (3 / (2↑𝑘)) |
| 45 | 41, 44 | eqtrdi 2793 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (2nd ‘(𝑀‘𝑘)) = (3 / (2↑𝑘))) |
| 46 | 45 | breq1d 5153 | . . . 4 ⊢ (𝑘 ∈ ℕ → ((2nd ‘(𝑀‘𝑘)) < 𝑟 ↔ (3 / (2↑𝑘)) < 𝑟)) |
| 47 | 46 | rexbiia 3092 | . . 3 ⊢ (∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟 ↔ ∃𝑘 ∈ ℕ (3 / (2↑𝑘)) < 𝑟) |
| 48 | 33, 47 | sylibr 234 | . 2 ⊢ (𝑟 ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟) |
| 49 | 48 | rgen 3063 | 1 ⊢ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 ⊆ wss 3951 ifcif 4525 𝒫 cpw 4600 〈cop 4632 ∪ cuni 4907 ∪ ciun 4991 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 2nd c2nd 8013 Fincfn 8985 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 3c3 12322 ℕ0cn0 12526 ℝ+crp 13034 seqcseq 14042 ↑cexp 14102 ballcbl 21351 MetOpencmopn 21354 CMetccmet 25288 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fl 13832 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: heiborlem8 37825 heiborlem9 37826 |
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