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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 36283. (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 12920 | . . 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 482 | . . 3 β’ ((π β§ π₯ β π) β π½ β Comp) |
7 | metxmet 23690 | . . . . . . 7 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
8 | 2, 7 | syl 17 | . . . . . 6 β’ (π β π· β (βMetβπ)) |
9 | 8 | adantr 482 | . . . . 5 β’ ((π β§ π₯ β π) β π· β (βMetβπ)) |
10 | 1 | mopntop 23796 | . . . . 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 486 | . . . . . 6 β’ ((π β§ π₯ β π) β π₯ β π) | |
13 | rpxr 12925 | . . . . . . 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 23774 | . . . . . 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 1372 | . . . . 5 β’ ((π β§ π₯ β π) β (π₯(ballβπ·)1) β π) |
17 | 1 | mopnuni 23797 | . . . . . 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 3985 | . . . 4 β’ ((π β§ π₯ β π) β (π₯(ballβπ·)1) β βͺ π½) |
20 | eqid 2737 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
21 | 20 | clscld 22401 | . . . 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 585 | . . 3 β’ ((π β§ π₯ β π) β ((clsβπ½)β(π₯(ballβπ·)1)) β (Clsdβπ½)) |
23 | cmpcld 22756 | . . 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 585 | . 2 β’ ((π β§ π₯ β π) β (π½ βΎt ((clsβπ½)β(π₯(ballβπ·)1))) β Comp) |
25 | 1, 2, 4, 24 | relcmpcmet 24685 | 1 β’ (π β π· β (CMetβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3911 βͺ cuni 4866 βcfv 6497 (class class class)co 7358 1c1 11053 β*cxr 11189 β+crp 12916 βΎt crest 17303 βMetcxmet 20784 Metcmet 20785 ballcbl 20786 MetOpencmopn 20789 Topctop 22245 Clsdccld 22370 clsccl 22372 Compccmp 22740 CMetccmet 24621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fi 9348 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-n0 12415 df-z 12501 df-uz 12765 df-q 12875 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-ico 13271 df-rest 17305 df-topgen 17326 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 df-mopn 20795 df-fbas 20796 df-fg 20797 df-top 22246 df-topon 22263 df-bases 22299 df-cld 22373 df-ntr 22374 df-cls 22375 df-nei 22452 df-cmp 22741 df-fil 23200 df-flim 23293 df-fcls 23295 df-cfil 24622 df-cmet 24624 |
This theorem is referenced by: (None) |
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