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Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem7 | Structured version Visualization version GIF version |
Description: Lemma for heibor 34489. 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 11513 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
2 | 3pos 11545 | . . . . . . 7 ⊢ 0 < 3 | |
3 | 1, 2 | elrpii 12200 | . . . . . 6 ⊢ 3 ∈ ℝ+ |
4 | rpdivcl 12224 | . . . . . 6 ⊢ ((𝑟 ∈ ℝ+ ∧ 3 ∈ ℝ+) → (𝑟 / 3) ∈ ℝ+) | |
5 | 3, 4 | mpan2 678 | . . . . 5 ⊢ (𝑟 ∈ ℝ+ → (𝑟 / 3) ∈ ℝ+) |
6 | 2re 11507 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 1lt2 11611 | . . . . . 6 ⊢ 1 < 2 | |
8 | expnlbnd 13402 | . . . . . 6 ⊢ (((𝑟 / 3) ∈ ℝ+ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑘 ∈ ℕ (1 / (2↑𝑘)) < (𝑟 / 3)) | |
9 | 6, 7, 8 | mp3an23 1432 | . . . . 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 11506 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ | |
12 | nnnn0 11708 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
13 | nnexpcl 13250 | . . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ) | |
14 | 11, 12, 13 | sylancr 578 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℕ) |
15 | 14 | nnrpd 12239 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℝ+) |
16 | rpcn 12209 | . . . . . . . . . 10 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ∈ ℂ) | |
17 | rpne0 12215 | . . . . . . . . . 10 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ≠ 0) | |
18 | 3cn 11514 | . . . . . . . . . . 11 ⊢ 3 ∈ ℂ | |
19 | divrec 11107 | . . . . . . . . . . 11 ⊢ ((3 ∈ ℂ ∧ (2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) | |
20 | 18, 19 | mp3an1 1427 | . . . . . . . . . 10 ⊢ (((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) |
21 | 16, 17, 20 | syl2anc 576 | . . . . . . . . 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 474 | . . . . . . 7 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) |
24 | 23 | breq1d 4933 | . . . . . 6 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → ((3 / (2↑𝑘)) < 𝑟 ↔ (3 · (1 / (2↑𝑘))) < 𝑟)) |
25 | 14 | nnrecred 11484 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ∈ ℝ) |
26 | rpre 12205 | . . . . . . 7 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ) | |
27 | 1, 2 | pm3.2i 463 | . . . . . . . 8 ⊢ (3 ∈ ℝ ∧ 0 < 3) |
28 | ltmuldiv2 11307 | . . . . . . . 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 1429 | . . . . . . 7 ⊢ (((1 / (2↑𝑘)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((3 · (1 / (2↑𝑘))) < 𝑟 ↔ (1 / (2↑𝑘)) < (𝑟 / 3))) |
30 | 25, 26, 29 | syl2anr 587 | . . . . . 6 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → ((3 · (1 / (2↑𝑘))) < 𝑟 ↔ (1 / (2↑𝑘)) < (𝑟 / 3))) |
31 | 24, 30 | bitrd 271 | . . . . 5 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → ((3 / (2↑𝑘)) < 𝑟 ↔ (1 / (2↑𝑘)) < (𝑟 / 3))) |
32 | 31 | rexbidva 3235 | . . . 4 ⊢ (𝑟 ∈ ℝ+ → (∃𝑘 ∈ ℕ (3 / (2↑𝑘)) < 𝑟 ↔ ∃𝑘 ∈ ℕ (1 / (2↑𝑘)) < (𝑟 / 3))) |
33 | 10, 32 | mpbird 249 | . . 3 ⊢ (𝑟 ∈ ℝ+ → ∃𝑘 ∈ ℕ (3 / (2↑𝑘)) < 𝑟) |
34 | fveq2 6493 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑆‘𝑛) = (𝑆‘𝑘)) | |
35 | oveq2 6978 | . . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘)) | |
36 | 35 | oveq2d 6986 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘))) |
37 | 34, 36 | opeq12d 4679 | . . . . . . . 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 5206 | . . . . . . . 8 ⊢ 〈(𝑆‘𝑘), (3 / (2↑𝑘))〉 ∈ V | |
40 | 37, 38, 39 | fvmpt 6589 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑀‘𝑘) = 〈(𝑆‘𝑘), (3 / (2↑𝑘))〉) |
41 | 40 | fveq2d 6497 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (2nd ‘(𝑀‘𝑘)) = (2nd ‘〈(𝑆‘𝑘), (3 / (2↑𝑘))〉)) |
42 | fvex 6506 | . . . . . . 7 ⊢ (𝑆‘𝑘) ∈ V | |
43 | ovex 7002 | . . . . . . 7 ⊢ (3 / (2↑𝑘)) ∈ V | |
44 | 42, 43 | op2nd 7503 | . . . . . 6 ⊢ (2nd ‘〈(𝑆‘𝑘), (3 / (2↑𝑘))〉) = (3 / (2↑𝑘)) |
45 | 41, 44 | syl6eq 2824 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (2nd ‘(𝑀‘𝑘)) = (3 / (2↑𝑘))) |
46 | 45 | breq1d 4933 | . . . 4 ⊢ (𝑘 ∈ ℕ → ((2nd ‘(𝑀‘𝑘)) < 𝑟 ↔ (3 / (2↑𝑘)) < 𝑟)) |
47 | 46 | rexbiia 3187 | . . 3 ⊢ (∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟 ↔ ∃𝑘 ∈ ℕ (3 / (2↑𝑘)) < 𝑟) |
48 | 33, 47 | sylibr 226 | . 2 ⊢ (𝑟 ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟) |
49 | 48 | rgen 3092 | 1 ⊢ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 {cab 2753 ≠ wne 2961 ∀wral 3082 ∃wrex 3083 ∩ cin 3824 ⊆ wss 3825 ifcif 4344 𝒫 cpw 4416 〈cop 4441 ∪ cuni 4706 ∪ ciun 4786 class class class wbr 4923 {copab 4985 ↦ cmpt 5002 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 2nd c2nd 7493 Fincfn 8298 ℂcc 10325 ℝcr 10326 0cc0 10327 1c1 10328 + caddc 10330 · cmul 10332 < clt 10466 − cmin 10662 / cdiv 11090 ℕcn 11431 2c2 11488 3c3 11489 ℕ0cn0 11700 ℝ+crp 12197 seqcseq 13177 ↑cexp 13237 ballcbl 20224 MetOpencmopn 20227 CMetccmet 23550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-z 11787 df-uz 12052 df-rp 12198 df-fl 12970 df-seq 13178 df-exp 13238 |
This theorem is referenced by: heiborlem8 34486 heiborlem9 34487 |
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