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| Mirrors > Home > MPE Home > Th. List > mopni3 | Structured version Visualization version GIF version | ||
| Description: An open set of a metric space includes an arbitrarily small ball around each of its points. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) |
| Ref | Expression |
|---|---|
| mopni.1 | β’ π½ = (MetOpenβπ·) |
| Ref | Expression |
|---|---|
| mopni3 | β’ (((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β§ π β β+) β βπ₯ β β+ (π₯ < π β§ (π(ballβπ·)π₯) β π΄)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mopni.1 | . . . 4 β’ π½ = (MetOpenβπ·) | |
| 2 | 1 | mopni2 24353 | . . 3 β’ ((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β βπ¦ β β+ (π(ballβπ·)π¦) β π΄) |
| 3 | 2 | adantr 480 | . 2 β’ (((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β§ π β β+) β βπ¦ β β+ (π(ballβπ·)π¦) β π΄) |
| 4 | simp1 1133 | . . . . . . 7 β’ ((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β π· β (βMetβπ)) | |
| 5 | 1 | mopnss 24303 | . . . . . . . . 9 β’ ((π· β (βMetβπ) β§ π΄ β π½) β π΄ β π) |
| 6 | 5 | sselda 3977 | . . . . . . . 8 β’ (((π· β (βMetβπ) β§ π΄ β π½) β§ π β π΄) β π β π) |
| 7 | 6 | 3impa 1107 | . . . . . . 7 β’ ((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β π β π) |
| 8 | 4, 7 | jca 511 | . . . . . 6 β’ ((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β (π· β (βMetβπ) β§ π β π)) |
| 9 | ssblex 24285 | . . . . . 6 β’ (((π· β (βMetβπ) β§ π β π) β§ (π β β+ β§ π¦ β β+)) β βπ₯ β β+ (π₯ < π β§ (π(ballβπ·)π₯) β (π(ballβπ·)π¦))) | |
| 10 | 8, 9 | sylan 579 | . . . . 5 β’ (((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β§ (π β β+ β§ π¦ β β+)) β βπ₯ β β+ (π₯ < π β§ (π(ballβπ·)π₯) β (π(ballβπ·)π¦))) |
| 11 | 10 | anassrs 467 | . . . 4 β’ ((((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β§ π β β+) β§ π¦ β β+) β βπ₯ β β+ (π₯ < π β§ (π(ballβπ·)π₯) β (π(ballβπ·)π¦))) |
| 12 | sstr 3985 | . . . . . . 7 β’ (((π(ballβπ·)π₯) β (π(ballβπ·)π¦) β§ (π(ballβπ·)π¦) β π΄) β (π(ballβπ·)π₯) β π΄) | |
| 13 | 12 | expcom 413 | . . . . . 6 β’ ((π(ballβπ·)π¦) β π΄ β ((π(ballβπ·)π₯) β (π(ballβπ·)π¦) β (π(ballβπ·)π₯) β π΄)) |
| 14 | 13 | anim2d 611 | . . . . 5 β’ ((π(ballβπ·)π¦) β π΄ β ((π₯ < π β§ (π(ballβπ·)π₯) β (π(ballβπ·)π¦)) β (π₯ < π β§ (π(ballβπ·)π₯) β π΄))) |
| 15 | 14 | reximdv 3164 | . . . 4 β’ ((π(ballβπ·)π¦) β π΄ β (βπ₯ β β+ (π₯ < π β§ (π(ballβπ·)π₯) β (π(ballβπ·)π¦)) β βπ₯ β β+ (π₯ < π β§ (π(ballβπ·)π₯) β π΄))) |
| 16 | 11, 15 | syl5com 31 | . . 3 β’ ((((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β§ π β β+) β§ π¦ β β+) β ((π(ballβπ·)π¦) β π΄ β βπ₯ β β+ (π₯ < π β§ (π(ballβπ·)π₯) β π΄))) |
| 17 | 16 | rexlimdva 3149 | . 2 β’ (((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β§ π β β+) β (βπ¦ β β+ (π(ballβπ·)π¦) β π΄ β βπ₯ β β+ (π₯ < π β§ (π(ballβπ·)π₯) β π΄))) |
| 18 | 3, 17 | mpd 15 | 1 β’ (((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β§ π β β+) β βπ₯ β β+ (π₯ < π β§ (π(ballβπ·)π₯) β π΄)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwrex 3064 β wss 3943 class class class wbr 5141 βcfv 6536 (class class class)co 7404 < clt 11249 β+crp 12977 βMetcxmet 21221 ballcbl 21223 MetOpencmopn 21226 |
| 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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-topgen 17396 df-psmet 21228 df-xmet 21229 df-bl 21231 df-mopn 21232 df-top 22747 df-topon 22764 df-bases 22800 |
| This theorem is referenced by: bcthlem5 25207 lhop1lem 25897 ulmdvlem3 26289 efopn 26543 opnrebl2 35714 |
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