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Theorem iscms 21516
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 5874 . . . 4  |-  ( Base `  w )  e.  _V
21a1i 11 . . 3  |-  ( w  =  M  ->  ( Base `  w )  e. 
_V )
3 fveq2 5864 . . . . . . 7  |-  ( w  =  M  ->  ( dist `  w )  =  ( dist `  M
) )
43adantr 465 . . . . . 6  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( dist `  w )  =  ( dist `  M
) )
5 id 22 . . . . . . . 8  |-  ( b  =  ( Base `  w
)  ->  b  =  ( Base `  w )
)
6 fveq2 5864 . . . . . . . . 9  |-  ( w  =  M  ->  ( Base `  w )  =  ( Base `  M
) )
7 iscms.1 . . . . . . . . 9  |-  X  =  ( Base `  M
)
86, 7syl6eqr 2526 . . . . . . . 8  |-  ( w  =  M  ->  ( Base `  w )  =  X )
95, 8sylan9eqr 2530 . . . . . . 7  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
b  =  X )
109, 9xpeq12d 5024 . . . . . 6  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( b  X.  b
)  =  ( X  X.  X ) )
114, 10reseq12d 5272 . . . . 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 2526 . . . 4  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( dist `  w
)  |`  ( b  X.  b ) )  =  D )
149fveq2d 5868 . . . 4  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( CMet `  b )  =  ( CMet `  X
) )
1513, 14eleq12d 2549 . . 3  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( ( dist `  w )  |`  (
b  X.  b ) )  e.  ( CMet `  b )  <->  D  e.  ( CMet `  X )
) )
162, 15sbcied 3368 . 2  |-  ( w  =  M  ->  ( [. ( Base `  w
)  /  b ]. ( ( dist `  w
)  |`  ( b  X.  b ) )  e.  ( CMet `  b
)  <->  D  e.  ( CMet `  X ) ) )
17 df-cms 21506 . 2  |- CMetSp  =  {
w  e.  MetSp  |  [. ( Base `  w )  /  b ]. (
( dist `  w )  |`  ( b  X.  b
) )  e.  (
CMet `  b ) }
1816, 17elrab2 3263 1  |-  ( M  e. CMetSp 
<->  ( M  e.  MetSp  /\  D  e.  ( CMet `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   [.wsbc 3331    X. cxp 4997    |` cres 5001   ` cfv 5586   Basecbs 14483   distcds 14557   MetSpcmt 20553   CMetcms 21425  CMetSpccms 21503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-res 5011  df-iota 5549  df-fv 5594  df-cms 21506
This theorem is referenced by:  cmscmet  21517  cmsms  21519  cmspropd  21520  cmsss  21521  cncms  21527
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