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Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem5 | Structured version Visualization version GIF version |
Description: Lemma for heibor 35673. The function 𝑀 is a set of point-and-radius pairs suitable for application to caubl 24177. (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 |
---|---|
heiborlem5 | ⊢ (𝜑 → 𝑀:ℕ⟶(𝑋 × ℝ+)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 12080 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
2 | inss1 4133 | . . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋 | |
3 | heibor.7 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) | |
4 | 3 | ffvelrnda 6893 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ (𝒫 𝑋 ∩ Fin)) |
5 | 2, 4 | sseldi 3889 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ 𝒫 𝑋) |
6 | 5 | elpwid 4514 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ⊆ 𝑋) |
7 | heibor.1 | . . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷) | |
8 | heibor.3 | . . . . . . . . 9 ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣} | |
9 | heibor.4 | . . . . . . . . 9 ⊢ 𝐺 = {〈𝑦, 𝑛〉 ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} | |
10 | heibor.5 | . . . . . . . . 9 ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) | |
11 | heibor.6 | . . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) | |
12 | heibor.8 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) | |
13 | heibor.9 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) | |
14 | heibor.10 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶𝐺0) | |
15 | heibor.11 | . . . . . . . . 9 ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) | |
16 | 7, 8, 9, 10, 11, 3, 12, 13, 14, 15 | heiborlem4 35666 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘)𝐺𝑘) |
17 | fvex 6719 | . . . . . . . . . 10 ⊢ (𝑆‘𝑘) ∈ V | |
18 | vex 3405 | . . . . . . . . . 10 ⊢ 𝑘 ∈ V | |
19 | 7, 8, 9, 17, 18 | heiborlem2 35664 | . . . . . . . . 9 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆‘𝑘) ∈ (𝐹‘𝑘) ∧ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾)) |
20 | 19 | simp2bi 1148 | . . . . . . . 8 ⊢ ((𝑆‘𝑘)𝐺𝑘 → (𝑆‘𝑘) ∈ (𝐹‘𝑘)) |
21 | 16, 20 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ (𝐹‘𝑘)) |
22 | 6, 21 | sseldd 3892 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ 𝑋) |
23 | 1, 22 | sylan2 596 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ 𝑋) |
24 | 23 | ralrimiva 3098 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ∈ 𝑋) |
25 | fveq2 6706 | . . . . . 6 ⊢ (𝑘 = 𝑛 → (𝑆‘𝑘) = (𝑆‘𝑛)) | |
26 | 25 | eleq1d 2818 | . . . . 5 ⊢ (𝑘 = 𝑛 → ((𝑆‘𝑘) ∈ 𝑋 ↔ (𝑆‘𝑛) ∈ 𝑋)) |
27 | 26 | cbvralvw 3351 | . . . 4 ⊢ (∀𝑘 ∈ ℕ (𝑆‘𝑘) ∈ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝑆‘𝑛) ∈ 𝑋) |
28 | 24, 27 | sylib 221 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑆‘𝑛) ∈ 𝑋) |
29 | 3re 11893 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
30 | 3pos 11918 | . . . . . . 7 ⊢ 0 < 3 | |
31 | 29, 30 | elrpii 12572 | . . . . . 6 ⊢ 3 ∈ ℝ+ |
32 | 2nn 11886 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
33 | nnnn0 12080 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
34 | nnexpcl 13631 | . . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ) | |
35 | 32, 33, 34 | sylancr 590 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ) |
36 | 35 | nnrpd 12609 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+) |
37 | rpdivcl 12594 | . . . . . 6 ⊢ ((3 ∈ ℝ+ ∧ (2↑𝑛) ∈ ℝ+) → (3 / (2↑𝑛)) ∈ ℝ+) | |
38 | 31, 36, 37 | sylancr 590 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (3 / (2↑𝑛)) ∈ ℝ+) |
39 | opelxpi 5577 | . . . . . 6 ⊢ (((𝑆‘𝑛) ∈ 𝑋 ∧ (3 / (2↑𝑛)) ∈ ℝ+) → 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+)) | |
40 | 39 | expcom 417 | . . . . 5 ⊢ ((3 / (2↑𝑛)) ∈ ℝ+ → ((𝑆‘𝑛) ∈ 𝑋 → 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+))) |
41 | 38, 40 | syl 17 | . . . 4 ⊢ (𝑛 ∈ ℕ → ((𝑆‘𝑛) ∈ 𝑋 → 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+))) |
42 | 41 | ralimia 3074 | . . 3 ⊢ (∀𝑛 ∈ ℕ (𝑆‘𝑛) ∈ 𝑋 → ∀𝑛 ∈ ℕ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+)) |
43 | 28, 42 | syl 17 | . 2 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+)) |
44 | heibor.12 | . . 3 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉) | |
45 | 44 | fmpt 6916 | . 2 ⊢ (∀𝑛 ∈ ℕ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+) ↔ 𝑀:ℕ⟶(𝑋 × ℝ+)) |
46 | 43, 45 | sylib 221 | 1 ⊢ (𝜑 → 𝑀:ℕ⟶(𝑋 × ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 {cab 2712 ∀wral 3054 ∃wrex 3055 ∩ cin 3856 ⊆ wss 3857 ifcif 4429 𝒫 cpw 4503 〈cop 4537 ∪ cuni 4809 ∪ ciun 4894 class class class wbr 5043 {copab 5105 ↦ cmpt 5124 × cxp 5538 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ∈ cmpo 7204 2nd c2nd 7749 Fincfn 8615 0cc0 10712 1c1 10713 + caddc 10715 − cmin 11045 / cdiv 11472 ℕcn 11813 2c2 11868 3c3 11869 ℕ0cn0 12073 ℝ+crp 12569 seqcseq 13557 ↑cexp 13618 ballcbl 20322 MetOpencmopn 20325 CMetccmet 24123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-seq 13558 df-exp 13619 |
This theorem is referenced by: heiborlem8 35670 heiborlem9 35671 |
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