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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 24001 | . . 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 481 | . 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 1136 | . . . . . . 7 β’ ((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β π· β (βMetβπ)) | |
| 5 | 1 | mopnss 23951 | . . . . . . . . 9 β’ ((π· β (βMetβπ) β§ π΄ β π½) β π΄ β π) |
| 6 | 5 | sselda 3982 | . . . . . . . 8 β’ (((π· β (βMetβπ) β§ π΄ β π½) β§ π β π΄) β π β π) |
| 7 | 6 | 3impa 1110 | . . . . . . 7 β’ ((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β π β π) |
| 8 | 4, 7 | jca 512 | . . . . . 6 β’ ((π· β (βMetβπ) β§ π΄ β π½ β§ π β π΄) β (π· β (βMetβπ) β§ π β π)) |
| 9 | ssblex 23933 | . . . . . 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 580 | . . . . 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 468 | . . . 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 3990 | . . . . . . 7 β’ (((π(ballβπ·)π₯) β (π(ballβπ·)π¦) β§ (π(ballβπ·)π¦) β π΄) β (π(ballβπ·)π₯) β π΄) | |
| 13 | 12 | expcom 414 | . . . . . 6 β’ ((π(ballβπ·)π¦) β π΄ β ((π(ballβπ·)π₯) β (π(ballβπ·)π¦) β (π(ballβπ·)π₯) β π΄)) |
| 14 | 13 | anim2d 612 | . . . . 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 3170 | . . . 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 3155 | . 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwrex 3070 β wss 3948 class class class wbr 5148 βcfv 6543 (class class class)co 7408 < clt 11247 β+crp 12973 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-top 22395 df-topon 22412 df-bases 22448 |
| This theorem is referenced by: bcthlem5 24844 lhop1lem 25529 ulmdvlem3 25913 efopn 26165 opnrebl2 35201 |
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