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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 25207 | . . . . 5 β’ (π· β (CMetβπ) β π· β (Metβπ)) | |
2 | metxmet 24233 | . . . . 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 25202 | . . . 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 4342 | . . 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 766 | . . 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 768 | . . 3 β’ (((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β§ π₯ β π) β πΉ β (Cauβπ·)) | |
12 | eqid 2728 | . . 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 25209 | . 2 β’ (((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β§ π₯ β π) β πΉ β dom (βπ‘βπ½)) |
14 | 7, 13 | exlimddv 1931 | 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 1534 βwex 1774 β wcel 2099 β wne 2936 β c0 4318 ifcif 4524 β¦ cmpt 5225 dom cdm 5672 βcfv 6542 βcn 12236 βMetcxmet 21257 Metcmet 21258 MetOpencmopn 21262 βπ‘clm 23123 Cauccau 25174 CMetccmet 25175 |
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 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 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-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-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-er 8718 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-inf 9460 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-ico 13356 df-rest 17397 df-topgen 17418 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-fbas 21269 df-fg 21270 df-top 22789 df-topon 22806 df-bases 22842 df-ntr 22917 df-nei 22995 df-lm 23126 df-fil 23743 df-fm 23835 df-flim 23836 df-flf 23837 df-cfil 25176 df-cau 25177 df-cmet 25178 |
This theorem is referenced by: iscmet3 25214 iscmet2 25215 bcthlem4 25248 minvecolem4a 30680 hlcompl 30718 heiborlem9 37286 bfplem1 37289 |
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