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Theorem ngpxms 20196
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 20195 . 2  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
2 msxms 20032 . 2  |-  ( G  e.  MetSp  ->  G  e.  *MetSp )
31, 2syl 16 1  |-  ( G  e. NrmGrp  ->  G  e.  *MetSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756   *MetSpcxme 19895   MetSpcmt 19896  NrmGrpcngp 20173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-rex 2724  df-rab 2727  df-v 2977  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-xp 4849  df-co 4852  df-res 4855  df-iota 5384  df-fv 5429  df-ms 19899  df-ngp 20179
This theorem is referenced by:  ngpdsr  20199  ngpds2r  20201  ngpds3  20202  ngpds3r  20203  nmge0  20211  nmeq0  20212  minveclem4a  20920  minveclem4  20922  qqhcn  26423  qqhucn  26424  rrhcn  26429  rrhf  26430  rrexttps  26438  rrexthaus  26439
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