![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13018 | . . 3 ⊢ (𝑅 ∈ ℝ+ → 𝑅 ∈ ℝ*) | |
2 | rpgt0 13021 | . . 3 ⊢ (𝑅 ∈ ℝ+ → 0 < 𝑅) | |
3 | 1, 2 | jca 510 | . 2 ⊢ (𝑅 ∈ ℝ+ → (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) |
4 | xblcntr 24361 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅)) | |
5 | 3, 4 | syl3an3 1162 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 0cc0 11140 ℝ*cxr 11279 < clt 11280 ℝ+crp 13009 ∞Metcxmet 21281 ballcbl 21283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-map 8847 df-xr 11284 df-rp 13010 df-psmet 21288 df-xmet 21289 df-bl 21291 |
This theorem is referenced by: bln0 24365 unirnbl 24370 blssex 24377 neibl 24454 blnei 24455 metss 24461 methaus 24473 met1stc 24474 met2ndci 24475 metrest 24477 prdsxmslem2 24482 metcnp3 24493 tgioo 24756 zdis 24776 metnrmlem2 24820 cnllycmp 24926 nmhmcn 25091 lmmbr 25230 cfilfcls 25246 iscmet3lem2 25264 caubl 25280 caublcls 25281 flimcfil 25286 ellimc3 25852 ulmdvlem1 26381 efopn 26637 logtayl 26639 xrlimcnp 26945 efrlim 26946 efrlimOLD 26947 lgamucov 27015 cnllysconn 34983 poimirlem30 37251 blbnd 37388 heibor1lem 37410 heibor1 37411 binomcxplemnotnn0 43932 hoiqssbl 46148 |
Copyright terms: Public domain | W3C validator |