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Theorem cmsms 19301
Description: A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
cmsms  |-  ( G  e. CMetSp  ->  G  e.  MetSp )

Proof of Theorem cmsms
StepHypRef Expression
1 eqid 2436 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2436 . . 3  |-  ( (
dist `  G )  |`  ( ( Base `  G
)  X.  ( Base `  G ) ) )  =  ( ( dist `  G )  |`  (
( Base `  G )  X.  ( Base `  G
) ) )
31, 2iscms 19298 . 2  |-  ( G  e. CMetSp 
<->  ( G  e.  MetSp  /\  ( ( dist `  G
)  |`  ( ( Base `  G )  X.  ( Base `  G ) ) )  e.  ( CMet `  ( Base `  G
) ) ) )
43simplbi 447 1  |-  ( G  e. CMetSp  ->  G  e.  MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    X. cxp 4876    |` cres 4880   ` cfv 5454   Basecbs 13469   distcds 13538   MetSpcmt 18348   CMetcms 19207  CMetSpccms 19285
This theorem is referenced by:  cmsss  19303  cmetcusp1OLD  19305  cmetcusp1  19306  rlmbn  19315  rrhre  24387  sitgclbn  24657
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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