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Mirrors > Home > MPE Home > Th. List > cmetcau | Structured version Visualization version GIF version |
Description: The convergence of a Cauchy sequence in a complete metric space. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Oct-2015.) |
Ref | Expression |
---|---|
cmetcau.1 | β’ π½ = (MetOpenβπ·) |
Ref | Expression |
---|---|
cmetcau | β’ ((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β πΉ β dom (βπ‘βπ½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmetmet 25138 | . . . . 5 β’ (π· β (CMetβπ) β π· β (Metβπ)) | |
2 | metxmet 24164 | . . . . 5 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
3 | 1, 2 | syl 17 | . . . 4 β’ (π· β (CMetβπ) β π· β (βMetβπ)) |
4 | caun0 25133 | . . . 4 β’ ((π· β (βMetβπ) β§ πΉ β (Cauβπ·)) β π β β ) | |
5 | 3, 4 | sylan 579 | . . 3 β’ ((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β π β β ) |
6 | n0 4339 | . . 3 β’ (π β β β βπ₯ π₯ β π) | |
7 | 5, 6 | sylib 217 | . 2 β’ ((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β βπ₯ π₯ β π) |
8 | cmetcau.1 | . . 3 β’ π½ = (MetOpenβπ·) | |
9 | simpll 764 | . . 3 β’ (((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β§ π₯ β π) β π· β (CMetβπ)) | |
10 | simpr 484 | . . 3 β’ (((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β§ π₯ β π) β π₯ β π) | |
11 | simplr 766 | . . 3 β’ (((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β§ π₯ β π) β πΉ β (Cauβπ·)) | |
12 | eqid 2724 | . . 3 β’ (π¦ β β β¦ if(π¦ β dom πΉ, (πΉβπ¦), π₯)) = (π¦ β β β¦ if(π¦ β dom πΉ, (πΉβπ¦), π₯)) | |
13 | 8, 9, 10, 11, 12 | cmetcaulem 25140 | . 2 β’ (((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β§ π₯ β π) β πΉ β dom (βπ‘βπ½)) |
14 | 7, 13 | exlimddv 1930 | 1 β’ ((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β πΉ β dom (βπ‘βπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 βwex 1773 β wcel 2098 β wne 2932 β c0 4315 ifcif 4521 β¦ cmpt 5222 dom cdm 5667 βcfv 6534 βcn 12210 βMetcxmet 21215 Metcmet 21216 MetOpencmopn 21220 βπ‘clm 23054 Cauccau 25105 CMetccmet 25106 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-n0 12471 df-z 12557 df-uz 12821 df-q 12931 df-rp 12973 df-xneg 13090 df-xadd 13091 df-xmul 13092 df-ico 13328 df-rest 17369 df-topgen 17390 df-psmet 21222 df-xmet 21223 df-met 21224 df-bl 21225 df-mopn 21226 df-fbas 21227 df-fg 21228 df-top 22720 df-topon 22737 df-bases 22773 df-ntr 22848 df-nei 22926 df-lm 23057 df-fil 23674 df-fm 23766 df-flim 23767 df-flf 23768 df-cfil 25107 df-cau 25108 df-cmet 25109 |
This theorem is referenced by: iscmet3 25145 iscmet2 25146 bcthlem4 25179 minvecolem4a 30602 hlcompl 30640 heiborlem9 37181 bfplem1 37184 |
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