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Theorem iscms 22313
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 5875 . . . 4  |-  ( Base `  w )  e.  _V
21a1i 11 . . 3  |-  ( w  =  M  ->  ( Base `  w )  e. 
_V )
3 fveq2 5865 . . . . . . 7  |-  ( w  =  M  ->  ( dist `  w )  =  ( dist `  M
) )
43adantr 467 . . . . . 6  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( dist `  w )  =  ( dist `  M
) )
5 id 22 . . . . . . . 8  |-  ( b  =  ( Base `  w
)  ->  b  =  ( Base `  w )
)
6 fveq2 5865 . . . . . . . . 9  |-  ( w  =  M  ->  ( Base `  w )  =  ( Base `  M
) )
7 iscms.1 . . . . . . . . 9  |-  X  =  ( Base `  M
)
86, 7syl6eqr 2503 . . . . . . . 8  |-  ( w  =  M  ->  ( Base `  w )  =  X )
95, 8sylan9eqr 2507 . . . . . . 7  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
b  =  X )
109sqxpeqd 4860 . . . . . 6  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( b  X.  b
)  =  ( X  X.  X ) )
114, 10reseq12d 5106 . . . . 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 2503 . . . 4  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( dist `  w
)  |`  ( b  X.  b ) )  =  D )
149fveq2d 5869 . . . 4  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( CMet `  b )  =  ( CMet `  X
) )
1513, 14eleq12d 2523 . . 3  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( ( dist `  w )  |`  (
b  X.  b ) )  e.  ( CMet `  b )  <->  D  e.  ( CMet `  X )
) )
162, 15sbcied 3304 . 2  |-  ( w  =  M  ->  ( [. ( Base `  w
)  /  b ]. ( ( dist `  w
)  |`  ( b  X.  b ) )  e.  ( CMet `  b
)  <->  D  e.  ( CMet `  X ) ) )
17 df-cms 22303 . 2  |- CMetSp  =  {
w  e.  MetSp  |  [. ( Base `  w )  /  b ]. (
( dist `  w )  |`  ( b  X.  b
) )  e.  (
CMet `  b ) }
1816, 17elrab2 3198 1  |-  ( M  e. CMetSp 
<->  ( M  e.  MetSp  /\  D  e.  ( CMet `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045   [.wsbc 3267    X. cxp 4832    |` cres 4836   ` cfv 5582   Basecbs 15121   distcds 15199   MetSpcmt 21333   CMetcms 22224  CMetSpccms 22300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-nul 4534
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-xp 4840  df-res 4846  df-iota 5546  df-fv 5590  df-cms 22303
This theorem is referenced by:  cmscmet  22314  cmsms  22316  cmspropd  22317  cmsss  22318  cncms  22322
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