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Theorem ngpxms 20152
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 20151 . 2  |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
2 msxms 19988 . 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 1761   *MetSpcxme 19851   MetSpcmt 19852  NrmGrpcngp 20129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-xp 4842  df-co 4845  df-res 4848  df-iota 5378  df-fv 5423  df-ms 19855  df-ngp 20135
This theorem is referenced by:  ngpdsr  20155  ngpds2r  20157  ngpds3  20158  ngpds3r  20159  nmge0  20167  nmeq0  20168  minveclem4a  20876  minveclem4  20878  qqhcn  26356  qqhucn  26357  rrhcn  26362  rrhf  26363  rrexttps  26371  rrexthaus  26372
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