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Theorem ngpxms 22215
Description: A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
ngpxms (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)

Proof of Theorem ngpxms
StepHypRef Expression
1 ngpms 22214 . 2 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2 msxms 22069 . 2 (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
31, 2syl 17 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  ∞MetSpcxme 21932  MetSpcmt 21933  NrmGrpcngp 22192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-co 5047  df-res 5050  df-iota 5768  df-fv 5812  df-ms 21936  df-ngp 22198
This theorem is referenced by:  ngpdsr  22219  ngpds2r  22221  ngpds3  22222  ngpds3r  22223  nmge0  22231  nmeq0  22232  minveclem4a  23009  minveclem4  23011  qqhcn  29363  qqhucn  29364  rrhcn  29369  rrhf  29370  rrexttps  29378  rrexthaus  29379
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