MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cmscmet Structured version   Unicode version

Theorem cmscmet 19291
Description: The induced metric on a complete normed group is complete. (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
cmscmet  |-  ( M  e. CMetSp  ->  D  e.  (
CMet `  X )
)

Proof of Theorem cmscmet
StepHypRef Expression
1 iscms.1 . . 3  |-  X  =  ( Base `  M
)
2 iscms.2 . . 3  |-  D  =  ( ( dist `  M
)  |`  ( X  X.  X ) )
31, 2iscms 19290 . 2  |-  ( M  e. CMetSp 
<->  ( M  e.  MetSp  /\  D  e.  ( CMet `  X ) ) )
43simprbi 451 1  |-  ( M  e. CMetSp  ->  D  e.  (
CMet `  X )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    X. cxp 4868    |` cres 4872   ` cfv 5446   Basecbs 13461   distcds 13530   MetSpcmt 18340   CMetcms 19199  CMetSpccms 19277
This theorem is referenced by:  bncmet  19292  cmsss  19295  cmetcusp1OLD  19297  cmetcusp1  19298  minveclem3a  19320
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 2416  ax-nul 4330
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 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-res 4882  df-iota 5410  df-fv 5454  df-cms 19280
  Copyright terms: Public domain W3C validator