Nearby theorems
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| Mirrors > Home > MPE Home > Th. List > blcls | Structured version Visualization version GIF version | ||
| Description: The closure of an open ball in a metric space is contained in the corresponding closed ball. (Equality need not hold; for example, with the discrete metric, the closed ball of radius 1 is the whole space, but the open ball of radius 1 is just a point, whose closure is also a point.) (Contributed by Mario Carneiro, 31-Dec-2013.) |
| Ref | Expression |
|---|---|
| mopni.1 | β’ π½ = (MetOpenβπ·) |
| blcld.3 | β’ π = {π§ β π β£ (ππ·π§) β€ π } |
| Ref | Expression |
|---|---|
| blcls | β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β ((clsβπ½)β(π(ballβπ·)π )) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mopni.1 | . . 3 β’ π½ = (MetOpenβπ·) | |
| 2 | blcld.3 | . . 3 β’ π = {π§ β π β£ (ππ·π§) β€ π } | |
| 3 | 1, 2 | blcld 24234 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β π β (Clsdβπ½)) |
| 4 | blssm 24144 | . . . 4 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β π) | |
| 5 | elbl 24114 | . . . . . 6 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π§ β (π(ballβπ·)π ) β (π§ β π β§ (ππ·π§) < π ))) | |
| 6 | xmetcl 24057 | . . . . . . . . . 10 β’ ((π· β (βMetβπ) β§ π β π β§ π§ β π) β (ππ·π§) β β*) | |
| 7 | 6 | 3expa 1118 | . . . . . . . . 9 β’ (((π· β (βMetβπ) β§ π β π) β§ π§ β π) β (ππ·π§) β β*) |
| 8 | 7 | 3adantl3 1168 | . . . . . . . 8 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π§ β π) β (ππ·π§) β β*) |
| 9 | simpl3 1193 | . . . . . . . 8 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π§ β π) β π β β*) | |
| 10 | xrltle 13132 | . . . . . . . 8 β’ (((ππ·π§) β β* β§ π β β*) β ((ππ·π§) < π β (ππ·π§) β€ π )) | |
| 11 | 8, 9, 10 | syl2anc 584 | . . . . . . 7 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π§ β π) β ((ππ·π§) < π β (ππ·π§) β€ π )) |
| 12 | 11 | expimpd 454 | . . . . . 6 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β ((π§ β π β§ (ππ·π§) < π ) β (ππ·π§) β€ π )) |
| 13 | 5, 12 | sylbid 239 | . . . . 5 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π§ β (π(ballβπ·)π ) β (ππ·π§) β€ π )) |
| 14 | 13 | ralrimiv 3145 | . . . 4 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β βπ§ β (π(ballβπ·)π )(ππ·π§) β€ π ) |
| 15 | ssrab 4070 | . . . 4 β’ ((π(ballβπ·)π ) β {π§ β π β£ (ππ·π§) β€ π } β ((π(ballβπ·)π ) β π β§ βπ§ β (π(ballβπ·)π )(ππ·π§) β€ π )) | |
| 16 | 4, 14, 15 | sylanbrc 583 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β {π§ β π β£ (ππ·π§) β€ π }) |
| 17 | 16, 2 | sseqtrrdi 4033 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) β π) |
| 18 | eqid 2732 | . . 3 β’ βͺ π½ = βͺ π½ | |
| 19 | 18 | clsss2 22796 | . 2 β’ ((π β (Clsdβπ½) β§ (π(ballβπ·)π ) β π) β ((clsβπ½)β(π(ballβπ·)π )) β π) |
| 20 | 3, 17, 19 | syl2anc 584 | 1 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β ((clsβπ½)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 βwral 3061 {crab 3432 β wss 3948 βͺ cuni 4908 class class class wbr 5148 βcfv 6543 (class class class)co 7411 β*cxr 11251 < clt 11252 β€ cle 11253 βMetcxmet 21129 ballcbl 21131 MetOpencmopn 21134 Clsdccld 22740 clsccl 22742 |
| 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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
| 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-int 4951 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-topgen 17393 df-psmet 21136 df-xmet 21137 df-bl 21139 df-mopn 21140 df-top 22616 df-topon 22633 df-bases 22669 df-cld 22743 df-cls 22745 |
| This theorem is referenced by: blsscls 24236 cnllycmp 24696 cncmet 25063 |
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