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Mirrors > Home > MPE Home > Th. List > cmetcvg | Structured version Visualization version GIF version |
Description: The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.) |
Ref | Expression |
---|---|
iscmet.1 | β’ π½ = (MetOpenβπ·) |
Ref | Expression |
---|---|
cmetcvg | β’ ((π· β (CMetβπ) β§ πΉ β (CauFilβπ·)) β (π½ fLim πΉ) β β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscmet.1 | . . . 4 β’ π½ = (MetOpenβπ·) | |
2 | 1 | iscmet 24671 | . . 3 β’ (π· β (CMetβπ) β (π· β (Metβπ) β§ βπ β (CauFilβπ·)(π½ fLim π) β β )) |
3 | 2 | simprbi 498 | . 2 β’ (π· β (CMetβπ) β βπ β (CauFilβπ·)(π½ fLim π) β β ) |
4 | oveq2 7369 | . . . 4 β’ (π = πΉ β (π½ fLim π) = (π½ fLim πΉ)) | |
5 | 4 | neeq1d 3000 | . . 3 β’ (π = πΉ β ((π½ fLim π) β β β (π½ fLim πΉ) β β )) |
6 | 5 | rspccva 3582 | . 2 β’ ((βπ β (CauFilβπ·)(π½ fLim π) β β β§ πΉ β (CauFilβπ·)) β (π½ fLim πΉ) β β ) |
7 | 3, 6 | sylan 581 | 1 β’ ((π· β (CMetβπ) β§ πΉ β (CauFilβπ·)) β (π½ fLim πΉ) β β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 β c0 4286 βcfv 6500 (class class class)co 7361 Metcmet 20805 MetOpencmopn 20809 fLim cflim 23308 CauFilccfil 24639 CMetccmet 24641 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-cmet 24644 |
This theorem is referenced by: cmetcaulem 24675 metsscmetcld 24702 cmetss 24703 cmetcusp 24741 minveclem4a 24817 fmcncfil 32576 |
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