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Theorem ngpxms 21246
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 21245 . 2  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
2 msxms 21082 . 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 1819   *MetSpcxme 20945   MetSpcmt 20946  NrmGrpcngp 21223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-co 5017  df-res 5020  df-iota 5557  df-fv 5602  df-ms 20949  df-ngp 21229
This theorem is referenced by:  ngpdsr  21249  ngpds2r  21251  ngpds3  21252  ngpds3r  21253  nmge0  21261  nmeq0  21262  minveclem4a  21970  minveclem4  21972  qqhcn  28125  qqhucn  28126  rrhcn  28131  rrhf  28132  rrexttps  28140  rrexthaus  28141
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