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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 24003 | . . 3 β’ (π· β (βMetβπ) β ran (ballβπ·) β π½) |
| 3 | 2 | 3ad2ant1 1133 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β ran (ballβπ·) β π½) |
| 4 | blelrn 23922 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β ran (ballβπ·)) | |
| 5 | 3, 4 | sseldd 3983 | 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 1087 = wceq 1541 β wcel 2106 β wss 3948 ran crn 5677 βcfv 6543 (class class class)co 7408 β*cxr 11246 βMetcxmet 20928 ballcbl 20930 MetOpencmopn 20933 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-topgen 17388 df-psmet 20935 df-xmet 20936 df-bl 20938 df-mopn 20939 df-bases 22448 |
| This theorem is referenced by: neibl 24009 blnei 24010 methaus 24028 met1stc 24029 met2ndci 24030 metrest 24032 prdsxmslem2 24037 metcnp3 24048 zdis 24331 metdseq0 24369 metnrmlem2 24375 cnheibor 24470 cnllycmp 24471 nmhmcn 24635 lmmbr 24774 cfilfcls 24790 iscmet3lem2 24808 flimcfil 24830 bcthlem5 24844 ellimc3 25395 dvlipcn 25510 dvlip2 25511 psercn 25937 pserdvlem2 25939 dvlog2 26160 efopnlem2 26164 logtayl 26167 xrlimcnp 26470 efrlim 26471 lgamucov 26539 cnllysconn 34231 poimirlem30 36513 heicant 36518 ismtyhmeolem 36667 heibor1lem 36672 heibor1 36673 binomcxplemdvbinom 43102 binomcxplemnotnn0 43105 ioorrnopnlem 45010 |
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