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| Mirrors > Home > MPE Home > Th. List > blopn | Structured version Visualization version GIF version | ||
| Description: A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) |
| Ref | Expression |
|---|---|
| mopni.1 | β’ π½ = (MetOpenβπ·) |
| Ref | Expression |
|---|---|
| blopn | β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β π½) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mopni.1 | . . . 4 β’ π½ = (MetOpenβπ·) | |
| 2 | 1 | blssopn 24403 | . . 3 β’ (π· β (βMetβπ) β ran (ballβπ·) β π½) |
| 3 | 2 | 3ad2ant1 1131 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β ran (ballβπ·) β π½) |
| 4 | blelrn 24322 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β ran (ballβπ·)) | |
| 5 | 3, 4 | sseldd 3981 | 1 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β π½) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 β wss 3947 ran crn 5679 βcfv 6548 (class class class)co 7420 β*cxr 11277 βMetcxmet 21263 ballcbl 21265 MetOpencmopn 21268 |
| 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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 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-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-topgen 17424 df-psmet 21270 df-xmet 21271 df-bl 21273 df-mopn 21274 df-bases 22848 |
| This theorem is referenced by: neibl 24409 blnei 24410 methaus 24428 met1stc 24429 met2ndci 24430 metrest 24432 prdsxmslem2 24437 metcnp3 24448 zdis 24731 metdseq0 24769 metnrmlem2 24775 cnheibor 24880 cnllycmp 24881 nmhmcn 25046 lmmbr 25185 cfilfcls 25201 iscmet3lem2 25219 flimcfil 25241 bcthlem5 25255 ellimc3 25807 dvlipcn 25926 dvlip2 25927 psercn 26362 pserdvlem2 26364 dvlog2 26586 efopnlem2 26590 logtayl 26593 xrlimcnp 26899 efrlim 26900 efrlimOLD 26901 lgamucov 26969 cnllysconn 34855 poimirlem30 37123 heicant 37128 ismtyhmeolem 37277 heibor1lem 37282 heibor1 37283 binomcxplemdvbinom 43790 binomcxplemnotnn0 43793 ioorrnopnlem 45692 |
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