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Mirrors > Home > MPE Home > Th. List > bcthlem3 | Structured version Visualization version GIF version |
Description: Lemma for bcth 25345. 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 7439 | . . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝐴 + 1))) | |
3 | id 22 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → 𝑘 = 𝐴) | |
4 | fveq2 6893 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (𝑔‘𝑘) = (𝑔‘𝐴)) | |
5 | 3, 4 | oveq12d 7434 | . . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝑘𝐹(𝑔‘𝑘)) = (𝐴𝐹(𝑔‘𝐴))) |
6 | 2, 5 | eleq12d 2820 | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴)))) |
7 | 6 | rspccva 3606 | . . . . . . 7 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝐴 ∈ ℕ) → (𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴))) |
8 | 1, 7 | sylan 578 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴))) |
9 | bcthlem.9 | . . . . . . . 8 ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+)) | |
10 | 9 | ffvelcdmda 7090 | . . . . . . 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 25340 | . . . . . . . 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 455 | . . . . . . 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 231 | . . . . 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 1141 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝐴)) ∖ (𝑀‘𝐴))) |
19 | 18 | difss2d 4131 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝐴))) |
20 | 19 | 3adant2 1128 | . 2 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝐴 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝐴))) |
21 | peano2nn 12270 | . . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
22 | cmetmet 25302 | . . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) | |
23 | metxmet 24328 | . . . . 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 25341 | . . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))) |
30 | 24, 9, 29, 11 | caublcls 25325 | . . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ (𝐴 + 1) ∈ ℕ) → 𝑥 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1))))) |
31 | 21, 30 | syl3an3 1162 | . 2 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝐴 ∈ ℕ) → 𝑥 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1))))) |
32 | 20, 31 | sseldd 3979 | 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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∖ cdif 3943 ⊆ wss 3946 〈cop 4629 class class class wbr 5145 {copab 5207 × cxp 5672 ∘ ccom 5678 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 ∈ cmpo 7418 1st c1st 7993 2nd c2nd 7994 1c1 11150 + caddc 11152 < clt 11289 / cdiv 11912 ℕcn 12258 ℝ+crp 13022 ∞Metcxmet 21324 Metcmet 21325 ballcbl 21326 MetOpencmopn 21329 Clsdccld 23008 clsccl 23010 ⇝𝑡clm 23218 CMetccmet 25270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-map 8849 df-pm 8850 df-en 8967 df-dom 8968 df-sdom 8969 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-n0 12519 df-z 12605 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-topgen 17453 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-top 22884 df-topon 22901 df-bases 22937 df-cld 23011 df-ntr 23012 df-cls 23013 df-lm 23221 df-cmet 25273 |
This theorem is referenced by: bcthlem4 25343 |
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