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Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem7 | Structured version Visualization version GIF version |
Description: Lemma for heibor 37808. 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 12344 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
2 | 3pos 12369 | . . . . . . 7 ⊢ 0 < 3 | |
3 | 1, 2 | elrpii 13035 | . . . . . 6 ⊢ 3 ∈ ℝ+ |
4 | rpdivcl 13058 | . . . . . 6 ⊢ ((𝑟 ∈ ℝ+ ∧ 3 ∈ ℝ+) → (𝑟 / 3) ∈ ℝ+) | |
5 | 3, 4 | mpan2 691 | . . . . 5 ⊢ (𝑟 ∈ ℝ+ → (𝑟 / 3) ∈ ℝ+) |
6 | 2re 12338 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 1lt2 12435 | . . . . . 6 ⊢ 1 < 2 | |
8 | expnlbnd 14269 | . . . . . 6 ⊢ (((𝑟 / 3) ∈ ℝ+ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑘 ∈ ℕ (1 / (2↑𝑘)) < (𝑟 / 3)) | |
9 | 6, 7, 8 | mp3an23 1452 | . . . . 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 12337 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ | |
12 | nnnn0 12531 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
13 | nnexpcl 14112 | . . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ) | |
14 | 11, 12, 13 | sylancr 587 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℕ) |
15 | 14 | nnrpd 13073 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℝ+) |
16 | rpcn 13043 | . . . . . . . . . 10 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ∈ ℂ) | |
17 | rpne0 13049 | . . . . . . . . . 10 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ≠ 0) | |
18 | 3cn 12345 | . . . . . . . . . . 11 ⊢ 3 ∈ ℂ | |
19 | divrec 11936 | . . . . . . . . . . 11 ⊢ ((3 ∈ ℂ ∧ (2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → (3 / (2↑𝑘)) = (3 · (1 / (2↑𝑘)))) | |
20 | 18, 19 | mp3an1 1447 | . . . . . . . . . 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 5158 | . . . . . 6 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ) → ((3 / (2↑𝑘)) < 𝑟 ↔ (3 · (1 / (2↑𝑘))) < 𝑟)) |
25 | 14 | nnrecred 12315 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ∈ ℝ) |
26 | rpre 13041 | . . . . . . 7 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ) | |
27 | 1, 2 | pm3.2i 470 | . . . . . . . 8 ⊢ (3 ∈ ℝ ∧ 0 < 3) |
28 | ltmuldiv2 12140 | . . . . . . . 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 1449 | . . . . . . 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 3175 | . . . 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 6907 | . . . . . . . . 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 4886 | . . . . . . . 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 5475 | . . . . . . . 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 6911 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → (2nd ‘(𝑀‘𝑘)) = (2nd ‘〈(𝑆‘𝑘), (3 / (2↑𝑘))〉)) |
42 | fvex 6920 | . . . . . . 7 ⊢ (𝑆‘𝑘) ∈ V | |
43 | ovex 7464 | . . . . . . 7 ⊢ (3 / (2↑𝑘)) ∈ V | |
44 | 42, 43 | op2nd 8022 | . . . . . 6 ⊢ (2nd ‘〈(𝑆‘𝑘), (3 / (2↑𝑘))〉) = (3 / (2↑𝑘)) |
45 | 41, 44 | eqtrdi 2791 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (2nd ‘(𝑀‘𝑘)) = (3 / (2↑𝑘))) |
46 | 45 | breq1d 5158 | . . . 4 ⊢ (𝑘 ∈ ℕ → ((2nd ‘(𝑀‘𝑘)) < 𝑟 ↔ (3 / (2↑𝑘)) < 𝑟)) |
47 | 46 | rexbiia 3090 | . . 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 3061 | 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 1086 = wceq 1537 ∈ wcel 2106 {cab 2712 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ∩ cin 3962 ⊆ wss 3963 ifcif 4531 𝒫 cpw 4605 〈cop 4637 ∪ cuni 4912 ∪ ciun 4996 class class class wbr 5148 {copab 5210 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 2nd c2nd 8012 Fincfn 8984 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 − cmin 11490 / cdiv 11918 ℕcn 12264 2c2 12319 3c3 12320 ℕ0cn0 12524 ℝ+crp 13032 seqcseq 14039 ↑cexp 14099 ballcbl 21369 MetOpencmopn 21372 CMetccmet 25302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fl 13829 df-seq 14040 df-exp 14100 |
This theorem is referenced by: heiborlem8 37805 heiborlem9 37806 |
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