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Theorem iscms 21968
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 5815 . . . 4  |-  ( Base `  w )  e.  _V
21a1i 11 . . 3  |-  ( w  =  M  ->  ( Base `  w )  e. 
_V )
3 fveq2 5805 . . . . . . 7  |-  ( w  =  M  ->  ( dist `  w )  =  ( dist `  M
) )
43adantr 463 . . . . . 6  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( dist `  w )  =  ( dist `  M
) )
5 id 22 . . . . . . . 8  |-  ( b  =  ( Base `  w
)  ->  b  =  ( Base `  w )
)
6 fveq2 5805 . . . . . . . . 9  |-  ( w  =  M  ->  ( Base `  w )  =  ( Base `  M
) )
7 iscms.1 . . . . . . . . 9  |-  X  =  ( Base `  M
)
86, 7syl6eqr 2461 . . . . . . . 8  |-  ( w  =  M  ->  ( Base `  w )  =  X )
95, 8sylan9eqr 2465 . . . . . . 7  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
b  =  X )
109sqxpeqd 4968 . . . . . 6  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( b  X.  b
)  =  ( X  X.  X ) )
114, 10reseq12d 5216 . . . . 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 2461 . . . 4  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( dist `  w
)  |`  ( b  X.  b ) )  =  D )
149fveq2d 5809 . . . 4  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( CMet `  b )  =  ( CMet `  X
) )
1513, 14eleq12d 2484 . . 3  |-  ( ( w  =  M  /\  b  =  ( Base `  w ) )  -> 
( ( ( dist `  w )  |`  (
b  X.  b ) )  e.  ( CMet `  b )  <->  D  e.  ( CMet `  X )
) )
162, 15sbcied 3313 . 2  |-  ( w  =  M  ->  ( [. ( Base `  w
)  /  b ]. ( ( dist `  w
)  |`  ( b  X.  b ) )  e.  ( CMet `  b
)  <->  D  e.  ( CMet `  X ) ) )
17 df-cms 21958 . 2  |- CMetSp  =  {
w  e.  MetSp  |  [. ( Base `  w )  /  b ]. (
( dist `  w )  |`  ( b  X.  b
) )  e.  (
CMet `  b ) }
1816, 17elrab2 3208 1  |-  ( M  e. CMetSp 
<->  ( M  e.  MetSp  /\  D  e.  ( CMet `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   [.wsbc 3276    X. cxp 4940    |` cres 4944   ` cfv 5525   Basecbs 14733   distcds 14810   MetSpcmt 21005   CMetcms 21877  CMetSpccms 21955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-xp 4948  df-res 4954  df-iota 5489  df-fv 5533  df-cms 21958
This theorem is referenced by:  cmscmet  21969  cmsms  21971  cmspropd  21972  cmsss  21973  cncms  21979
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