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Mirrors > Home > MPE Home > Th. List > bcthlem3 | Structured version Visualization version GIF version |
Description: Lemma for bcth 23496. The limit point of the centers in the sequence is in the intersection of every ball in the sequence. (Contributed by Mario Carneiro, 7-Jan-2014.) |
Ref | Expression |
---|---|
bcth.2 | ⊢ 𝐽 = (MetOpen‘𝐷) |
bcthlem.4 | ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) |
bcthlem.5 | ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {〈𝑥, 𝑟〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))}) |
bcthlem.6 | ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽)) |
bcthlem.7 | ⊢ (𝜑 → 𝑅 ∈ ℝ+) |
bcthlem.8 | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
bcthlem.9 | ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+)) |
bcthlem.10 | ⊢ (𝜑 → (𝑔‘1) = 〈𝐶, 𝑅〉) |
bcthlem.11 | ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))) |
Ref | Expression |
---|---|
bcthlem3 | ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝐴 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bcthlem.11 | . . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))) | |
2 | fvoveq1 6927 | . . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝐴 + 1))) | |
3 | id 22 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → 𝑘 = 𝐴) | |
4 | fveq2 6432 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (𝑔‘𝑘) = (𝑔‘𝐴)) | |
5 | 3, 4 | oveq12d 6922 | . . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝑘𝐹(𝑔‘𝑘)) = (𝐴𝐹(𝑔‘𝐴))) |
6 | 2, 5 | eleq12d 2899 | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴)))) |
7 | 6 | rspccva 3524 | . . . . . . 7 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝐴 ∈ ℕ) → (𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴))) |
8 | 1, 7 | sylan 577 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴))) |
9 | bcthlem.9 | . . . . . . . 8 ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+)) | |
10 | 9 | ffvelrnda 6607 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑔‘𝐴) ∈ (𝑋 × ℝ+)) |
11 | bcth.2 | . . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷) | |
12 | bcthlem.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) | |
13 | bcthlem.5 | . . . . . . . . 9 ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {〈𝑥, 𝑟〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))}) | |
14 | 11, 12, 13 | bcthlem1 23491 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ (𝑔‘𝐴) ∈ (𝑋 × ℝ+))) → ((𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴)) ↔ ((𝑔‘(𝐴 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝐴 + 1))) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝐴)) ∖ (𝑀‘𝐴))))) |
15 | 14 | expr 450 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((𝑔‘𝐴) ∈ (𝑋 × ℝ+) → ((𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴)) ↔ ((𝑔‘(𝐴 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝐴 + 1))) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝐴)) ∖ (𝑀‘𝐴)))))) |
16 | 10, 15 | mpd 15 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴)) ↔ ((𝑔‘(𝐴 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝐴 + 1))) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝐴)) ∖ (𝑀‘𝐴))))) |
17 | 8, 16 | mpbid 224 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((𝑔‘(𝐴 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝐴 + 1))) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝐴)) ∖ (𝑀‘𝐴)))) |
18 | 17 | simp3d 1180 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝐴)) ∖ (𝑀‘𝐴))) |
19 | 18 | difss2d 3966 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝐴))) |
20 | 19 | 3adant2 1167 | . 2 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝐴 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝐴))) |
21 | peano2nn 11363 | . . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
22 | cmetmet 23453 | . . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) | |
23 | metxmet 22508 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
24 | 12, 22, 23 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
25 | bcthlem.6 | . . . . 5 ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽)) | |
26 | bcthlem.7 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
27 | bcthlem.8 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
28 | bcthlem.10 | . . . . 5 ⊢ (𝜑 → (𝑔‘1) = 〈𝐶, 𝑅〉) | |
29 | 11, 12, 13, 25, 26, 27, 9, 28, 1 | bcthlem2 23492 | . . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))) |
30 | 24, 9, 29, 11 | caublcls 23476 | . . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ (𝐴 + 1) ∈ ℕ) → 𝑥 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1))))) |
31 | 21, 30 | syl3an3 1211 | . 2 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝐴 ∈ ℕ) → 𝑥 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1))))) |
32 | 20, 31 | sseldd 3827 | 1 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝐴 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3116 ∖ cdif 3794 ⊆ wss 3797 〈cop 4402 class class class wbr 4872 {copab 4934 × cxp 5339 ∘ ccom 5345 ⟶wf 6118 ‘cfv 6122 (class class class)co 6904 ↦ cmpt2 6906 1st c1st 7425 2nd c2nd 7426 1c1 10252 + caddc 10254 < clt 10390 / cdiv 11008 ℕcn 11349 ℝ+crp 12111 ∞Metcxmet 20090 Metcmet 20091 ballcbl 20092 MetOpencmopn 20095 Clsdccld 21190 clsccl 21192 ⇝𝑡clm 21400 CMetccmet 23421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-map 8123 df-pm 8124 df-en 8222 df-dom 8223 df-sdom 8224 df-sup 8616 df-inf 8617 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-n0 11618 df-z 11704 df-uz 11968 df-q 12071 df-rp 12112 df-xneg 12231 df-xadd 12232 df-xmul 12233 df-topgen 16456 df-psmet 20097 df-xmet 20098 df-met 20099 df-bl 20100 df-mopn 20101 df-top 21068 df-topon 21085 df-bases 21120 df-cld 21193 df-ntr 21194 df-cls 21195 df-lm 21403 df-cmet 23424 |
This theorem is referenced by: bcthlem4 23494 |
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