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Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem7 | Structured version Visualization version GIF version |
Description: Lemma for heibor 36280. 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 12233 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
2 | 3pos 12258 | . . . . . . 7 ⊢ 0 < 3 | |
3 | 1, 2 | elrpii 12918 | . . . . . 6 ⊢ 3 ∈ ℝ+ |
4 | rpdivcl 12940 | . . . . . 6 ⊢ ((𝑟 ∈ ℝ+ ∧ 3 ∈ ℝ+) → (𝑟 / 3) ∈ ℝ+) | |
5 | 3, 4 | mpan2 689 | . . . . 5 ⊢ (𝑟 ∈ ℝ+ → (𝑟 / 3) ∈ ℝ+) |
6 | 2re 12227 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 1lt2 12324 | . . . . . 6 ⊢ 1 < 2 | |
8 | expnlbnd 14136 | . . . . . 6 ⊢ (((𝑟 / 3) ∈ ℝ+ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑘 ∈ ℕ (1 / (2↑𝑘)) < (𝑟 / 3)) | |
9 | 6, 7, 8 | mp3an23 1453 | . . . . 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 12226 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ | |
12 | nnnn0 12420 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
13 | nnexpcl 13980 | . . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ) | |
14 | 11, 12, 13 | sylancr 587 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℕ) |
15 | 14 | nnrpd 12955 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℝ+) |
16 | rpcn 12925 | . . . . . . . . . 10 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ∈ ℂ) | |
17 | rpne0 12931 | . . . . . . . . . 10 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ≠ 0) | |
18 | 3cn 12234 | . . . . . . . . . . 11 ⊢ 3 ∈ ℂ | |
19 | divrec 11829 | . . . . . . . . . . 11 ⊢ ((3 ∈ ℂ ∧ (2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) | |
20 | 18, 19 | mp3an1 1448 | . . . . . . . . . 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 482 | . . . . . . 7 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) |
24 | 23 | breq1d 5115 | . . . . . 6 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → ((3 / (2↑𝑘)) < 𝑟 ↔ (3 · (1 / (2↑𝑘))) < 𝑟)) |
25 | 14 | nnrecred 12204 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ∈ ℝ) |
26 | rpre 12923 | . . . . . . 7 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ) | |
27 | 1, 2 | pm3.2i 471 | . . . . . . . 8 ⊢ (3 ∈ ℝ ∧ 0 < 3) |
28 | ltmuldiv2 12029 | . . . . . . . 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 1450 | . . . . . . 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 278 | . . . . 5 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → ((3 / (2↑𝑘)) < 𝑟 ↔ (1 / (2↑𝑘)) < (𝑟 / 3))) |
32 | 31 | rexbidva 3173 | . . . 4 ⊢ (𝑟 ∈ ℝ+ → (∃𝑘 ∈ ℕ (3 / (2↑𝑘)) < 𝑟 ↔ ∃𝑘 ∈ ℕ (1 / (2↑𝑘)) < (𝑟 / 3))) |
33 | 10, 32 | mpbird 256 | . . 3 ⊢ (𝑟 ∈ ℝ+ → ∃𝑘 ∈ ℕ (3 / (2↑𝑘)) < 𝑟) |
34 | fveq2 6842 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑆‘𝑛) = (𝑆‘𝑘)) | |
35 | oveq2 7365 | . . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘)) | |
36 | 35 | oveq2d 7373 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘))) |
37 | 34, 36 | opeq12d 4838 | . . . . . . . 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 5421 | . . . . . . . 8 ⊢ 〈(𝑆‘𝑘), (3 / (2↑𝑘))〉 ∈ V | |
40 | 37, 38, 39 | fvmpt 6948 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑀‘𝑘) = 〈(𝑆‘𝑘), (3 / (2↑𝑘))〉) |
41 | 40 | fveq2d 6846 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (2nd ‘(𝑀‘𝑘)) = (2nd ‘〈(𝑆‘𝑘), (3 / (2↑𝑘))〉)) |
42 | fvex 6855 | . . . . . . 7 ⊢ (𝑆‘𝑘) ∈ V | |
43 | ovex 7390 | . . . . . . 7 ⊢ (3 / (2↑𝑘)) ∈ V | |
44 | 42, 43 | op2nd 7930 | . . . . . 6 ⊢ (2nd ‘〈(𝑆‘𝑘), (3 / (2↑𝑘))〉) = (3 / (2↑𝑘)) |
45 | 41, 44 | eqtrdi 2792 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (2nd ‘(𝑀‘𝑘)) = (3 / (2↑𝑘))) |
46 | 45 | breq1d 5115 | . . . 4 ⊢ (𝑘 ∈ ℕ → ((2nd ‘(𝑀‘𝑘)) < 𝑟 ↔ (3 / (2↑𝑘)) < 𝑟)) |
47 | 46 | rexbiia 3095 | . . 3 ⊢ (∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟 ↔ ∃𝑘 ∈ ℕ (3 / (2↑𝑘)) < 𝑟) |
48 | 33, 47 | sylibr 233 | . 2 ⊢ (𝑟 ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟) |
49 | 48 | rgen 3066 | 1 ⊢ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {cab 2713 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 ∩ cin 3909 ⊆ wss 3910 ifcif 4486 𝒫 cpw 4560 〈cop 4592 ∪ cuni 4865 ∪ ciun 4954 class class class wbr 5105 {copab 5167 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 2nd c2nd 7920 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 < clt 11189 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 3c3 12209 ℕ0cn0 12413 ℝ+crp 12915 seqcseq 13906 ↑cexp 13967 ballcbl 20783 MetOpencmopn 20786 CMetccmet 24618 |
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-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 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 |
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-op 4593 df-uni 4866 df-iun 4956 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-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-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 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-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fl 13697 df-seq 13907 df-exp 13968 |
This theorem is referenced by: heiborlem8 36277 heiborlem9 36278 |
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