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Theorem ngpxms 22776
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 22775 . 2 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2 msxms 22630 . 2 (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
31, 2syl 17 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166  ∞MetSpcxms 22493  MetSpcms 22494  NrmGrpcngp 22753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-xp 5349  df-co 5352  df-res 5355  df-iota 6087  df-fv 6132  df-ms 22497  df-ngp 22759
This theorem is referenced by:  ngpdsr  22780  ngpds2r  22782  ngpds3  22783  ngpds3r  22784  nmge0  22792  nmeq0  22793  minveclem4a  23599  minveclem4  23601  qqhcn  30581  qqhucn  30582  rrhcn  30587  rrhf  30588  rrexttps  30596  rrexthaus  30597
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