Nearby theorems
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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 24348 | . . 3 β’ (π· β (βMetβπ) β ran (ballβπ·) β π½) |
| 3 | 2 | 3ad2ant1 1130 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β ran (ballβπ·) β π½) |
| 4 | blelrn 24267 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β ran (ballβπ·)) | |
| 5 | 3, 4 | sseldd 3976 | 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 1084 = wceq 1533 β wcel 2098 β wss 3941 ran crn 5668 βcfv 6534 (class class class)co 7402 β*cxr 11246 βMetcxmet 21219 ballcbl 21221 MetOpencmopn 21224 |
| 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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 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 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-topgen 17394 df-psmet 21226 df-xmet 21227 df-bl 21229 df-mopn 21230 df-bases 22793 |
| This theorem is referenced by: neibl 24354 blnei 24355 methaus 24373 met1stc 24374 met2ndci 24375 metrest 24377 prdsxmslem2 24382 metcnp3 24393 zdis 24676 metdseq0 24714 metnrmlem2 24720 cnheibor 24825 cnllycmp 24826 nmhmcn 24991 lmmbr 25130 cfilfcls 25146 iscmet3lem2 25164 flimcfil 25186 bcthlem5 25200 ellimc3 25752 dvlipcn 25871 dvlip2 25872 psercn 26303 pserdvlem2 26305 dvlog2 26527 efopnlem2 26531 logtayl 26534 xrlimcnp 26840 efrlim 26841 efrlimOLD 26842 lgamucov 26910 cnllysconn 34753 poimirlem30 37021 heicant 37026 ismtyhmeolem 37175 heibor1lem 37180 heibor1 37181 binomcxplemdvbinom 43661 binomcxplemnotnn0 43664 ioorrnopnlem 45565 |
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