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Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem5 | Structured version Visualization version GIF version |
Description: Lemma for heibor 35101. The function 𝑀 is a set of point-and-radius pairs suitable for application to caubl 23913. (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 11907 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
2 | inss1 4207 | . . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋 | |
3 | heibor.7 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) | |
4 | 3 | ffvelrnda 6853 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ (𝒫 𝑋 ∩ Fin)) |
5 | 2, 4 | sseldi 3967 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ 𝒫 𝑋) |
6 | 5 | elpwid 4552 | . . . . . . 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 35094 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘)𝐺𝑘) |
17 | fvex 6685 | . . . . . . . . . 10 ⊢ (𝑆‘𝑘) ∈ V | |
18 | vex 3499 | . . . . . . . . . 10 ⊢ 𝑘 ∈ V | |
19 | 7, 8, 9, 17, 18 | heiborlem2 35092 | . . . . . . . . 9 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆‘𝑘) ∈ (𝐹‘𝑘) ∧ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾)) |
20 | 19 | simp2bi 1142 | . . . . . . . 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 3970 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ 𝑋) |
23 | 1, 22 | sylan2 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ 𝑋) |
24 | 23 | ralrimiva 3184 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ∈ 𝑋) |
25 | fveq2 6672 | . . . . . 6 ⊢ (𝑘 = 𝑛 → (𝑆‘𝑘) = (𝑆‘𝑛)) | |
26 | 25 | eleq1d 2899 | . . . . 5 ⊢ (𝑘 = 𝑛 → ((𝑆‘𝑘) ∈ 𝑋 ↔ (𝑆‘𝑛) ∈ 𝑋)) |
27 | 26 | cbvralvw 3451 | . . . 4 ⊢ (∀𝑘 ∈ ℕ (𝑆‘𝑘) ∈ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝑆‘𝑛) ∈ 𝑋) |
28 | 24, 27 | sylib 220 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑆‘𝑛) ∈ 𝑋) |
29 | 3re 11720 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
30 | 3pos 11745 | . . . . . . 7 ⊢ 0 < 3 | |
31 | 29, 30 | elrpii 12395 | . . . . . 6 ⊢ 3 ∈ ℝ+ |
32 | 2nn 11713 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
33 | nnnn0 11907 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
34 | nnexpcl 13445 | . . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ) | |
35 | 32, 33, 34 | sylancr 589 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ) |
36 | 35 | nnrpd 12432 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+) |
37 | rpdivcl 12417 | . . . . . 6 ⊢ ((3 ∈ ℝ+ ∧ (2↑𝑛) ∈ ℝ+) → (3 / (2↑𝑛)) ∈ ℝ+) | |
38 | 31, 36, 37 | sylancr 589 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (3 / (2↑𝑛)) ∈ ℝ+) |
39 | opelxpi 5594 | . . . . . 6 ⊢ (((𝑆‘𝑛) ∈ 𝑋 ∧ (3 / (2↑𝑛)) ∈ ℝ+) → 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+)) | |
40 | 39 | expcom 416 | . . . . 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 3160 | . . 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 6876 | . 2 ⊢ (∀𝑛 ∈ ℕ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+) ↔ 𝑀:ℕ⟶(𝑋 × ℝ+)) |
46 | 43, 45 | sylib 220 | 1 ⊢ (𝜑 → 𝑀:ℕ⟶(𝑋 × ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 ⊆ wss 3938 ifcif 4469 𝒫 cpw 4541 〈cop 4575 ∪ cuni 4840 ∪ ciun 4921 class class class wbr 5068 {copab 5130 ↦ cmpt 5148 × cxp 5555 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 2nd c2nd 7690 Fincfn 8511 0cc0 10539 1c1 10540 + caddc 10542 − cmin 10872 / cdiv 11299 ℕcn 11640 2c2 11695 3c3 11696 ℕ0cn0 11900 ℝ+crp 12392 seqcseq 13372 ↑cexp 13432 ballcbl 20534 MetOpencmopn 20537 CMetccmet 23859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 |
This theorem is referenced by: heiborlem8 35098 heiborlem9 35099 |
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