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Mirrors > Home > MPE Home > Th. List > bcthlem3 | Structured version Visualization version GIF version |
Description: Lemma for bcth 23625. 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 6993 | . . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝐴 + 1))) | |
3 | id 22 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → 𝑘 = 𝐴) | |
4 | fveq2 6493 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (𝑔‘𝑘) = (𝑔‘𝐴)) | |
5 | 3, 4 | oveq12d 6988 | . . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝑘𝐹(𝑔‘𝑘)) = (𝐴𝐹(𝑔‘𝐴))) |
6 | 2, 5 | eleq12d 2854 | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴)))) |
7 | 6 | rspccva 3528 | . . . . . . 7 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝐴 ∈ ℕ) → (𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴))) |
8 | 1, 7 | sylan 572 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝑔‘(𝐴 + 1)) ∈ (𝐴𝐹(𝑔‘𝐴))) |
9 | bcthlem.9 | . . . . . . . 8 ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+)) | |
10 | 9 | ffvelrnda 6670 | . . . . . . 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 23620 | . . . . . . . 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 449 | . . . . . . 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 1124 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝐴)) ∖ (𝑀‘𝐴))) |
19 | 18 | difss2d 3997 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝐴))) |
20 | 19 | 3adant2 1111 | . 2 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝐴 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝐴))) |
21 | peano2nn 11445 | . . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
22 | cmetmet 23582 | . . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) | |
23 | metxmet 22637 | . . . . 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 23621 | . . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))) |
30 | 24, 9, 29, 11 | caublcls 23605 | . . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ (𝐴 + 1) ∈ ℕ) → 𝑥 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1))))) |
31 | 21, 30 | syl3an3 1145 | . 2 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝐴 ∈ ℕ) → 𝑥 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝐴 + 1))))) |
32 | 20, 31 | sseldd 3855 | 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 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ∀wral 3082 ∖ cdif 3822 ⊆ wss 3825 〈cop 4441 class class class wbr 4923 {copab 4985 × cxp 5398 ∘ ccom 5404 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 1st c1st 7492 2nd c2nd 7493 1c1 10328 + caddc 10330 < clt 10466 / cdiv 11090 ℕcn 11431 ℝ+crp 12197 ∞Metcxmet 20222 Metcmet 20223 ballcbl 20224 MetOpencmopn 20227 Clsdccld 21318 clsccl 21320 ⇝𝑡clm 21528 CMetccmet 23550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-map 8200 df-pm 8201 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-n0 11701 df-z 11787 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-topgen 16563 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-top 21196 df-topon 21213 df-bases 21248 df-cld 21321 df-ntr 21322 df-cls 21323 df-lm 21531 df-cmet 23553 |
This theorem is referenced by: bcthlem4 23623 |
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