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Theorem ngpxms 23980
Description: A normed group is an extended 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 23979 . 2 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2 msxms 23830 . 2 (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
31, 2syl 17 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  ∞MetSpcxms 23693  MetSpcms 23694  NrmGrpcngp 23956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-co 5646  df-res 5649  df-iota 6452  df-fv 6508  df-ms 23697  df-ngp 23962
This theorem is referenced by:  ngpdsr  23984  ngpds2r  23986  ngpds3  23987  ngpds3r  23988  nmge0  23996  nmeq0  23997  minveclem4a  24817  minveclem4  24819  qqhcn  32636  qqhucn  32637  rrhcn  32642  rrhf  32643  rrexttps  32651  rrexthaus  32652
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