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Theorem iscms 18767
Description: A complete metric space is a metric space with a complete metric. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
iscms.1  |-  X  =  ( Base `  M
)
iscms.2  |-  D  =  ( ( dist `  M
)  |`  ( X  X.  X ) )
Assertion
Ref Expression
iscms  |-  ( M  e. CMetSp 
<->  ( M  e.  MetSp  /\  D  e.  ( CMet `  X ) ) )

Proof of Theorem iscms
Dummy variables  w  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5539 . . . 4  |-  ( Base `  w )  e.  _V
21a1i 10 . . 3  |-  ( w  =  M  ->  ( Base `  w )  e. 
_V )
3 fveq2 5525 . . . . . . 7  |-  ( w  =  M  ->  ( dist `  w )  =  ( dist `  M
) )
43adantr 451 . . . . . 6  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( dist `  w )  =  ( dist `  M
) )
5 id 19 . . . . . . . 8  |-  ( b  =  ( Base `  w
)  ->  b  =  ( Base `  w )
)
6 fveq2 5525 . . . . . . . . 9  |-  ( w  =  M  ->  ( Base `  w )  =  ( Base `  M
) )
7 iscms.1 . . . . . . . . 9  |-  X  =  ( Base `  M
)
86, 7syl6eqr 2333 . . . . . . . 8  |-  ( w  =  M  ->  ( Base `  w )  =  X )
95, 8sylan9eqr 2337 . . . . . . 7  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
b  =  X )
109, 9xpeq12d 4714 . . . . . 6  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( b  X.  b
)  =  ( X  X.  X ) )
114, 10reseq12d 4956 . . . . 5  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( dist `  w
)  |`  ( b  X.  b ) )  =  ( ( dist `  M
)  |`  ( X  X.  X ) ) )
12 iscms.2 . . . . 5  |-  D  =  ( ( dist `  M
)  |`  ( X  X.  X ) )
1311, 12syl6eqr 2333 . . . 4  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( dist `  w
)  |`  ( b  X.  b ) )  =  D )
149fveq2d 5529 . . . 4  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( CMet `  b )  =  ( CMet `  X
) )
1513, 14eleq12d 2351 . . 3  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( ( dist `  w )  |`  (
b  X.  b ) )  e.  ( CMet `  b )  <->  D  e.  ( CMet `  X )
) )
162, 15sbcied 3027 . 2  |-  ( w  =  M  ->  ( [. ( Base `  w
)  /  b ]. ( ( dist `  w
)  |`  ( b  X.  b ) )  e.  ( CMet `  b
)  <->  D  e.  ( CMet `  X ) ) )
17 df-cms 18757 . 2  |- CMetSp  =  {
w  e.  MetSp  |  [. ( Base `  w )  /  b ]. (
( dist `  w )  |`  ( b  X.  b
) )  e.  (
CMet `  b ) }
1816, 17elrab2 2925 1  |-  ( M  e. CMetSp 
<->  ( M  e.  MetSp  /\  D  e.  ( CMet `  X ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991    X. cxp 4687    |` cres 4691   ` cfv 5255   Basecbs 13148   distcds 13217   MetSpcmt 17883   CMetcms 18680  CMetSpccms 18754
This theorem is referenced by:  cmscmet  18768  cmsms  18770  cmspropd  18771  cmsss  18772  cncms  18774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-res 4701  df-iota 5219  df-fv 5263  df-cms 18757
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