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Mirrors > Home > MPE Home > Th. List > blcntr | Structured version Visualization version GIF version |
Description: A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
blcntr | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpxr 11878 | . . 3 ⊢ (𝑅 ∈ ℝ+ → 𝑅 ∈ ℝ*) | |
2 | rpgt0 11882 | . . 3 ⊢ (𝑅 ∈ ℝ+ → 0 < 𝑅) | |
3 | 1, 2 | jca 553 | . 2 ⊢ (𝑅 ∈ ℝ+ → (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) |
4 | xblcntr 22263 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅)) | |
5 | 3, 4 | syl3an3 1401 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 0cc0 9974 ℝ*cxr 10111 < clt 10112 ℝ+crp 11870 ∞Metcxmt 19779 ballcbl 19781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-map 7901 df-xr 10116 df-rp 11871 df-psmet 19786 df-xmet 19787 df-bl 19789 |
This theorem is referenced by: bln0 22267 unirnbl 22272 blssex 22279 neibl 22353 blnei 22354 metss 22360 methaus 22372 met1stc 22373 met2ndci 22374 metrest 22376 prdsxmslem2 22381 metcnp3 22392 tgioo 22646 zdis 22666 metnrmlem2 22710 cnllycmp 22802 nmhmcn 22966 lmmbr 23102 cfilfcls 23118 iscmet3lem2 23136 caubl 23152 caublcls 23153 flimcfil 23158 ellimc3 23688 ulmdvlem1 24199 efopn 24449 logtayl 24451 xrlimcnp 24740 efrlim 24741 lgamucov 24809 cnllysconn 31353 poimirlem30 33569 blbnd 33716 heibor1lem 33738 heibor1 33739 binomcxplemnotnn0 38872 hoiqssbl 41160 |
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