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Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem5 | Structured version Visualization version GIF version |
Description: Lemma for heibor 33620. The function 𝑀 is a set of point-and-radius pairs suitable for application to caubl 23106. (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 11299 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
2 | inss1 3833 | . . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋 | |
3 | heibor.7 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) | |
4 | 3 | ffvelrnda 6359 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ (𝒫 𝑋 ∩ Fin)) |
5 | 2, 4 | sseldi 3601 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ 𝒫 𝑋) |
6 | 5 | elpwid 4170 | . . . . . . 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 33613 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘)𝐺𝑘) |
17 | fvex 6201 | . . . . . . . . . 10 ⊢ (𝑆‘𝑘) ∈ V | |
18 | vex 3203 | . . . . . . . . . 10 ⊢ 𝑘 ∈ V | |
19 | 7, 8, 9, 17, 18 | heiborlem2 33611 | . . . . . . . . 9 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆‘𝑘) ∈ (𝐹‘𝑘) ∧ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾)) |
20 | 19 | simp2bi 1077 | . . . . . . . 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 3604 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ 𝑋) |
23 | 1, 22 | sylan2 491 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ 𝑋) |
24 | 23 | ralrimiva 2966 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ∈ 𝑋) |
25 | fveq2 6191 | . . . . . 6 ⊢ (𝑘 = 𝑛 → (𝑆‘𝑘) = (𝑆‘𝑛)) | |
26 | 25 | eleq1d 2686 | . . . . 5 ⊢ (𝑘 = 𝑛 → ((𝑆‘𝑘) ∈ 𝑋 ↔ (𝑆‘𝑛) ∈ 𝑋)) |
27 | 26 | cbvralv 3171 | . . . 4 ⊢ (∀𝑘 ∈ ℕ (𝑆‘𝑘) ∈ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝑆‘𝑛) ∈ 𝑋) |
28 | 24, 27 | sylib 208 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑆‘𝑛) ∈ 𝑋) |
29 | 3re 11094 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
30 | 3pos 11114 | . . . . . . 7 ⊢ 0 < 3 | |
31 | 29, 30 | elrpii 11835 | . . . . . 6 ⊢ 3 ∈ ℝ+ |
32 | 2nn 11185 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
33 | nnnn0 11299 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
34 | nnexpcl 12873 | . . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ) | |
35 | 32, 33, 34 | sylancr 695 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ) |
36 | 35 | nnrpd 11870 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+) |
37 | rpdivcl 11856 | . . . . . 6 ⊢ ((3 ∈ ℝ+ ∧ (2↑𝑛) ∈ ℝ+) → (3 / (2↑𝑛)) ∈ ℝ+) | |
38 | 31, 36, 37 | sylancr 695 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (3 / (2↑𝑛)) ∈ ℝ+) |
39 | opelxpi 5148 | . . . . . 6 ⊢ (((𝑆‘𝑛) ∈ 𝑋 ∧ (3 / (2↑𝑛)) ∈ ℝ+) → 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+)) | |
40 | 39 | expcom 451 | . . . . 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 2950 | . . 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 6381 | . 2 ⊢ (∀𝑛 ∈ ℕ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉 ∈ (𝑋 × ℝ+) ↔ 𝑀:ℕ⟶(𝑋 × ℝ+)) |
46 | 43, 45 | sylib 208 | 1 ⊢ (𝜑 → 𝑀:ℕ⟶(𝑋 × ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {cab 2608 ∀wral 2912 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 ifcif 4086 𝒫 cpw 4158 〈cop 4183 ∪ cuni 4436 ∪ ciun 4520 class class class wbr 4653 {copab 4712 ↦ cmpt 4729 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 2nd c2nd 7167 Fincfn 7955 0cc0 9936 1c1 9937 + caddc 9939 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 3c3 11071 ℕ0cn0 11292 ℝ+crp 11832 seqcseq 12801 ↑cexp 12860 ballcbl 19733 MetOpencmopn 19736 CMetcms 23052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 |
This theorem is referenced by: heiborlem8 33617 heiborlem9 33618 |
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