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Mirrors > Home > MPE Home > Th. List > cmpcmet | Structured version Visualization version GIF version |
Description: A compact metric space is complete. One half of heibor 37202. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
relcmpcmet.1 | β’ π½ = (MetOpenβπ·) |
relcmpcmet.2 | β’ (π β π· β (Metβπ)) |
cmpcmet.3 | β’ (π β π½ β Comp) |
Ref | Expression |
---|---|
cmpcmet | β’ (π β π· β (CMetβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcmpcmet.1 | . 2 β’ π½ = (MetOpenβπ·) | |
2 | relcmpcmet.2 | . 2 β’ (π β π· β (Metβπ)) | |
3 | 1rp 12984 | . . 3 β’ 1 β β+ | |
4 | 3 | a1i 11 | . 2 β’ (π β 1 β β+) |
5 | cmpcmet.3 | . . . 4 β’ (π β π½ β Comp) | |
6 | 5 | adantr 480 | . . 3 β’ ((π β§ π₯ β π) β π½ β Comp) |
7 | metxmet 24195 | . . . . . . 7 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
8 | 2, 7 | syl 17 | . . . . . 6 β’ (π β π· β (βMetβπ)) |
9 | 8 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β π) β π· β (βMetβπ)) |
10 | 1 | mopntop 24301 | . . . . 5 β’ (π· β (βMetβπ) β π½ β Top) |
11 | 9, 10 | syl 17 | . . . 4 β’ ((π β§ π₯ β π) β π½ β Top) |
12 | simpr 484 | . . . . . 6 β’ ((π β§ π₯ β π) β π₯ β π) | |
13 | rpxr 12989 | . . . . . . 7 β’ (1 β β+ β 1 β β*) | |
14 | 3, 13 | mp1i 13 | . . . . . 6 β’ ((π β§ π₯ β π) β 1 β β*) |
15 | blssm 24279 | . . . . . 6 β’ ((π· β (βMetβπ) β§ π₯ β π β§ 1 β β*) β (π₯(ballβπ·)1) β π) | |
16 | 9, 12, 14, 15 | syl3anc 1368 | . . . . 5 β’ ((π β§ π₯ β π) β (π₯(ballβπ·)1) β π) |
17 | 1 | mopnuni 24302 | . . . . . 6 β’ (π· β (βMetβπ) β π = βͺ π½) |
18 | 9, 17 | syl 17 | . . . . 5 β’ ((π β§ π₯ β π) β π = βͺ π½) |
19 | 16, 18 | sseqtrd 4017 | . . . 4 β’ ((π β§ π₯ β π) β (π₯(ballβπ·)1) β βͺ π½) |
20 | eqid 2726 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
21 | 20 | clscld 22906 | . . . 4 β’ ((π½ β Top β§ (π₯(ballβπ·)1) β βͺ π½) β ((clsβπ½)β(π₯(ballβπ·)1)) β (Clsdβπ½)) |
22 | 11, 19, 21 | syl2anc 583 | . . 3 β’ ((π β§ π₯ β π) β ((clsβπ½)β(π₯(ballβπ·)1)) β (Clsdβπ½)) |
23 | cmpcld 23261 | . . 3 β’ ((π½ β Comp β§ ((clsβπ½)β(π₯(ballβπ·)1)) β (Clsdβπ½)) β (π½ βΎt ((clsβπ½)β(π₯(ballβπ·)1))) β Comp) | |
24 | 6, 22, 23 | syl2anc 583 | . 2 β’ ((π β§ π₯ β π) β (π½ βΎt ((clsβπ½)β(π₯(ballβπ·)1))) β Comp) |
25 | 1, 2, 4, 24 | relcmpcmet 25201 | 1 β’ (π β π· β (CMetβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3943 βͺ cuni 4902 βcfv 6537 (class class class)co 7405 1c1 11113 β*cxr 11251 β+crp 12980 βΎt crest 17375 βMetcxmet 21225 Metcmet 21226 ballcbl 21227 MetOpencmopn 21230 Topctop 22750 Clsdccld 22875 clsccl 22877 Compccmp 23245 CMetccmet 25137 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 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-int 4944 df-iun 4992 df-iin 4993 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 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 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ico 13336 df-rest 17377 df-topgen 17398 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-top 22751 df-topon 22768 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-cmp 23246 df-fil 23705 df-flim 23798 df-fcls 23800 df-cfil 25138 df-cmet 25140 |
This theorem is referenced by: (None) |
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