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Theorem iscms 19298
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 5742 . . . 4  |-  ( Base `  w )  e.  _V
21a1i 11 . . 3  |-  ( w  =  M  ->  ( Base `  w )  e. 
_V )
3 fveq2 5728 . . . . . . 7  |-  ( w  =  M  ->  ( dist `  w )  =  ( dist `  M
) )
43adantr 452 . . . . . 6  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( dist `  w )  =  ( dist `  M
) )
5 id 20 . . . . . . . 8  |-  ( b  =  ( Base `  w
)  ->  b  =  ( Base `  w )
)
6 fveq2 5728 . . . . . . . . 9  |-  ( w  =  M  ->  ( Base `  w )  =  ( Base `  M
) )
7 iscms.1 . . . . . . . . 9  |-  X  =  ( Base `  M
)
86, 7syl6eqr 2486 . . . . . . . 8  |-  ( w  =  M  ->  ( Base `  w )  =  X )
95, 8sylan9eqr 2490 . . . . . . 7  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
b  =  X )
109, 9xpeq12d 4903 . . . . . 6  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( b  X.  b
)  =  ( X  X.  X ) )
114, 10reseq12d 5147 . . . . 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 2486 . . . 4  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( dist `  w
)  |`  ( b  X.  b ) )  =  D )
149fveq2d 5732 . . . 4  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( CMet `  b )  =  ( CMet `  X
) )
1513, 14eleq12d 2504 . . 3  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( ( dist `  w )  |`  (
b  X.  b ) )  e.  ( CMet `  b )  <->  D  e.  ( CMet `  X )
) )
162, 15sbcied 3197 . 2  |-  ( w  =  M  ->  ( [. ( Base `  w
)  /  b ]. ( ( dist `  w
)  |`  ( b  X.  b ) )  e.  ( CMet `  b
)  <->  D  e.  ( CMet `  X ) ) )
17 df-cms 19288 . 2  |- CMetSp  =  {
w  e.  MetSp  |  [. ( Base `  w )  /  b ]. (
( dist `  w )  |`  ( b  X.  b
) )  e.  (
CMet `  b ) }
1816, 17elrab2 3094 1  |-  ( M  e. CMetSp 
<->  ( M  e.  MetSp  /\  D  e.  ( CMet `  X ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   [.wsbc 3161    X. cxp 4876    |` cres 4880   ` cfv 5454   Basecbs 13469   distcds 13538   MetSpcmt 18348   CMetcms 19207  CMetSpccms 19285
This theorem is referenced by:  cmscmet  19299  cmsms  19301  cmspropd  19302  cmsss  19303  cncms  19309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-res 4890  df-iota 5418  df-fv 5462  df-cms 19288
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