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Theorem cmetmet 18728
Description: A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 29-Jan-2014.)
Assertion
Ref Expression
cmetmet  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )

Proof of Theorem cmetmet
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . 3  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
21iscmet 18726 . 2  |-  ( D  e.  ( CMet `  X
)  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( ( MetOpen `  D
)  fLim  f )  =/=  (/) ) )
32simplbi 446 1  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    =/= wne 2459   A.wral 2556   (/)c0 3468   ` cfv 5271  (class class class)co 5874   Metcme 16386   MetOpencmopn 16388    fLim cflim 17645  CauFilccfil 18694   CMetcms 18696
This theorem is referenced by:  cmetmeti  18729  cmetcaulem  18730  cmetcau  18731  iscmet2  18736  cmetss  18756  bcthlem2  18763  bcthlem3  18764  bcthlem4  18765  bcthlem5  18766  bcth2  18768  bcth3  18769  minveclem3  18809  ubthlem1  21465  ubthlem2  21466  hlmet  21490  heiborlem3  26640  heiborlem6  26643  heiborlem8  26645  heiborlem9  26646  heiborlem10  26647  heibor  26648  bfplem1  26649  bfplem2  26650  bfp  26651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-cmet 18699
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