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Theorem ngpxms 18123
Description: A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
ngpxms  |-  ( G  e. NrmGrp  ->  G  e.  * MetSp )

Proof of Theorem ngpxms
StepHypRef Expression
1 ngpms 18122 . 2  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
2 msxms 18000 . 2  |-  ( G  e.  MetSp  ->  G  e.  *
MetSp )
31, 2syl 15 1  |-  ( G  e. NrmGrp  ->  G  e.  * MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   * MetSpcxme 17882   MetSpcmt 17883  NrmGrpcngp 18100
This theorem is referenced by:  ngpdsr  18126  ngpds2r  18128  ngpds3  18129  ngpds3r  18130  nmge0  18138  nmeq0  18139  minveclem4a  18794  minveclem4  18796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-co 4698  df-res 4701  df-iota 5219  df-fv 5263  df-ms 17886  df-ngp 18106
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