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Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem5 | Structured version Visualization version GIF version |
Description: Lemma for heibor 36280. The function 𝑀 is a set of point-and-radius pairs suitable for application to caubl 24672. (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 12420 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
2 | inss1 4188 | . . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋 | |
3 | heibor.7 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) | |
4 | 3 | ffvelcdmda 7035 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ (𝒫 𝑋 ∩ Fin)) |
5 | 2, 4 | sselid 3942 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ 𝒫 𝑋) |
6 | 5 | elpwid 4569 | . . . . . . 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 36273 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘)𝐺𝑘) |
17 | fvex 6855 | . . . . . . . . . 10 ⊢ (𝑆‘𝑘) ∈ V | |
18 | vex 3449 | . . . . . . . . . 10 ⊢ 𝑘 ∈ V | |
19 | 7, 8, 9, 17, 18 | heiborlem2 36271 | . . . . . . . . 9 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆‘𝑘) ∈ (𝐹‘𝑘) ∧ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾)) |
20 | 19 | simp2bi 1146 | . . . . . . . 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 3945 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ 𝑋) |
23 | 1, 22 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ 𝑋) |
24 | 23 | ralrimiva 3143 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ∈ 𝑋) |
25 | fveq2 6842 | . . . . . 6 ⊢ (𝑘 = 𝑛 → (𝑆‘𝑘) = (𝑆‘𝑛)) | |
26 | 25 | eleq1d 2822 | . . . . 5 ⊢ (𝑘 = 𝑛 → ((𝑆‘𝑘) ∈ 𝑋 ↔ (𝑆‘𝑛) ∈ 𝑋)) |
27 | 26 | cbvralvw 3225 | . . . 4 ⊢ (∀𝑘 ∈ ℕ (𝑆‘𝑘) ∈ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝑆‘𝑛) ∈ 𝑋) |
28 | 24, 27 | sylib 217 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑆‘𝑛) ∈ 𝑋) |
29 | 3re 12233 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
30 | 3pos 12258 | . . . . . . 7 ⊢ 0 < 3 | |
31 | 29, 30 | elrpii 12918 | . . . . . 6 ⊢ 3 ∈ ℝ+ |
32 | 2nn 12226 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
33 | nnnn0 12420 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
34 | nnexpcl 13980 | . . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ) | |
35 | 32, 33, 34 | sylancr 587 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ) |
36 | 35 | nnrpd 12955 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+) |
37 | rpdivcl 12940 | . . . . . 6 ⊢ ((3 ∈ ℝ+ ∧ (2↑𝑛) ∈ ℝ+) → (3 / (2↑𝑛)) ∈ ℝ+) | |
38 | 31, 36, 37 | sylancr 587 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (3 / (2↑𝑛)) ∈ ℝ+) |
39 | opelxpi 5670 | . . . . . 6 ⊢ (((𝑆‘𝑛) ∈ 𝑋 ∧ (3 / (2↑𝑛)) ∈ ℝ+) → 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+)) | |
40 | 39 | expcom 414 | . . . . 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 3083 | . . 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 7058 | . 2 ⊢ (∀𝑛 ∈ ℕ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+) ↔ 𝑀:ℕ⟶(𝑋 × ℝ+)) |
46 | 43, 45 | sylib 217 | 1 ⊢ (𝜑 → 𝑀:ℕ⟶(𝑋 × ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {cab 2713 ∀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 × cxp 5631 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 2nd c2nd 7920 Fincfn 8883 0cc0 11051 1c1 11052 + caddc 11054 − 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 |
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-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-seq 13907 df-exp 13968 |
This theorem is referenced by: heiborlem8 36277 heiborlem9 36278 |
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