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Mirrors > Home > MPE Home > Th. List > metcld | Structured version Visualization version GIF version |
Description: A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of [Kreyszig] p. 30. (Contributed by NM, 11-Nov-2007.) (Revised by Mario Carneiro, 1-May-2014.) |
Ref | Expression |
---|---|
metcld.2 | β’ π½ = (MetOpenβπ·) |
Ref | Expression |
---|---|
metcld | β’ ((π· β (βMetβπ) β§ π β π) β (π β (Clsdβπ½) β βπ₯βπ((π:ββΆπ β§ π(βπ‘βπ½)π₯) β π₯ β π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metcld.2 | . . . 4 β’ π½ = (MetOpenβπ·) | |
2 | 1 | mopntop 24339 | . . 3 β’ (π· β (βMetβπ) β π½ β Top) |
3 | 1 | mopnuni 24340 | . . . . 5 β’ (π· β (βMetβπ) β π = βͺ π½) |
4 | 3 | sseq2d 4010 | . . . 4 β’ (π· β (βMetβπ) β (π β π β π β βͺ π½)) |
5 | 4 | biimpa 476 | . . 3 β’ ((π· β (βMetβπ) β§ π β π) β π β βͺ π½) |
6 | eqid 2727 | . . . 4 β’ βͺ π½ = βͺ π½ | |
7 | 6 | iscld4 22962 | . . 3 β’ ((π½ β Top β§ π β βͺ π½) β (π β (Clsdβπ½) β ((clsβπ½)βπ) β π)) |
8 | 2, 5, 7 | syl2an2r 684 | . 2 β’ ((π· β (βMetβπ) β§ π β π) β (π β (Clsdβπ½) β ((clsβπ½)βπ) β π)) |
9 | 19.23v 1938 | . . . . 5 β’ (βπ((π:ββΆπ β§ π(βπ‘βπ½)π₯) β π₯ β π) β (βπ(π:ββΆπ β§ π(βπ‘βπ½)π₯) β π₯ β π)) | |
10 | simpl 482 | . . . . . . 7 β’ ((π· β (βMetβπ) β§ π β π) β π· β (βMetβπ)) | |
11 | simpr 484 | . . . . . . 7 β’ ((π· β (βMetβπ) β§ π β π) β π β π) | |
12 | 1, 10, 11 | metelcls 25226 | . . . . . 6 β’ ((π· β (βMetβπ) β§ π β π) β (π₯ β ((clsβπ½)βπ) β βπ(π:ββΆπ β§ π(βπ‘βπ½)π₯))) |
13 | 12 | imbi1d 341 | . . . . 5 β’ ((π· β (βMetβπ) β§ π β π) β ((π₯ β ((clsβπ½)βπ) β π₯ β π) β (βπ(π:ββΆπ β§ π(βπ‘βπ½)π₯) β π₯ β π))) |
14 | 9, 13 | bitr4id 290 | . . . 4 β’ ((π· β (βMetβπ) β§ π β π) β (βπ((π:ββΆπ β§ π(βπ‘βπ½)π₯) β π₯ β π) β (π₯ β ((clsβπ½)βπ) β π₯ β π))) |
15 | 14 | albidv 1916 | . . 3 β’ ((π· β (βMetβπ) β§ π β π) β (βπ₯βπ((π:ββΆπ β§ π(βπ‘βπ½)π₯) β π₯ β π) β βπ₯(π₯ β ((clsβπ½)βπ) β π₯ β π))) |
16 | dfss2 3964 | . . 3 β’ (((clsβπ½)βπ) β π β βπ₯(π₯ β ((clsβπ½)βπ) β π₯ β π)) | |
17 | 15, 16 | bitr4di 289 | . 2 β’ ((π· β (βMetβπ) β§ π β π) β (βπ₯βπ((π:ββΆπ β§ π(βπ‘βπ½)π₯) β π₯ β π) β ((clsβπ½)βπ) β π)) |
18 | 8, 17 | bitr4d 282 | 1 β’ ((π· β (βMetβπ) β§ π β π) β (π β (Clsdβπ½) β βπ₯βπ((π:ββΆπ β§ π(βπ‘βπ½)π₯) β π₯ β π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 βwal 1532 = wceq 1534 βwex 1774 β wcel 2099 β wss 3944 βͺ cuni 4903 class class class wbr 5142 βΆwf 6538 βcfv 6542 βcn 12236 βMetcxmet 21257 MetOpencmopn 21262 Topctop 22788 Clsdccld 22913 clsccl 22915 βπ‘clm 23123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cc 10452 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-card 9956 df-acn 9959 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-n0 12497 df-z 12583 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-fz 13511 df-topgen 17418 df-psmet 21264 df-xmet 21265 df-bl 21267 df-mopn 21268 df-top 22789 df-topon 22806 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-lm 23126 df-1stc 23336 |
This theorem is referenced by: metcld2 25228 |
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